L 2 ndash Vectors and Scalars

Outlinebull Physical quantities - vectors and scalarsbull Addition and subtraction of vectorbull Resultant vectorbull Change in a vector quantity calculating relative

and resolve a vector into componentsbull Vector representation in a component form in a

coordinate system

Vectors in Physics-Examples

Many physical quantities have both magnitude and direction they are called vectorsbullExamples displacement velocity acceleration force momentum

Other physical quantities have only magnitude they are called scalars bullExamples distance speed mass energy

Displacement and Distance

bull Displacement is the vector connecting a starting point A and some final point B

A

B

bull Distance is the length one would travel from point A to the final point B Therefore distance is a scalar

Geometrical Representation of Vectors

bull Arrows on a plane or spacebull To indicate a vector we use bold letters or an arrow on top of a letter

Properties of Vectors

1 The opposite of a vector a is vector - a

2 It has the same length but opposite direction

3 Two vectors a and b are parallel if one is a positive multiple of the other

a = m b mgt0

Example

if a = 3 b then a is parallel to b

(if a = -2 b then a is anti-parallel to b)

Operations Adding two Vectors

When we add two vectors we get the resultant vector a + b with the parallelogram rule

Operations Adding more vectors

bull We can add more vectors by pairing themappropriately

Operations Vector Subtraction

bull Special case of vector addition Add the negative of the subtracted

vectorbull a ndash b = a + (ndash b)

Components of a Vector

bull A component is a part or shadow along a given direction

bull It is useful to use rectangular componentsndash These are the

projections of the vector along the x- and y-axes

Components of a Vector cont

bull The x-component of a vector is the projection along the x-axis

ax = a cosθ

bull The y-component of a vector is the projection along the y-axis

ay = a sinθ

bull a is the magnitude of vector a a2 = ax

2 + ay2

Example 1

Resolve this vector along the x and y axes to find its components respectively

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Vectors in Physics-Examples

Many physical quantities have both magnitude and direction they are called vectorsbullExamples displacement velocity acceleration force momentum

Other physical quantities have only magnitude they are called scalars bullExamples distance speed mass energy

Displacement and Distance

bull Displacement is the vector connecting a starting point A and some final point B

A

B

bull Distance is the length one would travel from point A to the final point B Therefore distance is a scalar

Geometrical Representation of Vectors

bull Arrows on a plane or spacebull To indicate a vector we use bold letters or an arrow on top of a letter

Properties of Vectors

1 The opposite of a vector a is vector - a

2 It has the same length but opposite direction

3 Two vectors a and b are parallel if one is a positive multiple of the other

a = m b mgt0

Example

if a = 3 b then a is parallel to b

(if a = -2 b then a is anti-parallel to b)

Operations Adding two Vectors

When we add two vectors we get the resultant vector a + b with the parallelogram rule

Operations Adding more vectors

bull We can add more vectors by pairing themappropriately

Operations Vector Subtraction

bull Special case of vector addition Add the negative of the subtracted

vectorbull a ndash b = a + (ndash b)

Components of a Vector

bull A component is a part or shadow along a given direction

bull It is useful to use rectangular componentsndash These are the

projections of the vector along the x- and y-axes

Components of a Vector cont

bull The x-component of a vector is the projection along the x-axis

ax = a cosθ

bull The y-component of a vector is the projection along the y-axis

ay = a sinθ

bull a is the magnitude of vector a a2 = ax

2 + ay2

Example 1

Resolve this vector along the x and y axes to find its components respectively

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Displacement and Distance

bull Displacement is the vector connecting a starting point A and some final point B

A

B

bull Distance is the length one would travel from point A to the final point B Therefore distance is a scalar

Geometrical Representation of Vectors

bull Arrows on a plane or spacebull To indicate a vector we use bold letters or an arrow on top of a letter

Properties of Vectors

1 The opposite of a vector a is vector - a

2 It has the same length but opposite direction

3 Two vectors a and b are parallel if one is a positive multiple of the other

a = m b mgt0

Example

if a = 3 b then a is parallel to b

(if a = -2 b then a is anti-parallel to b)

Operations Adding two Vectors

When we add two vectors we get the resultant vector a + b with the parallelogram rule

Operations Adding more vectors

bull We can add more vectors by pairing themappropriately

Operations Vector Subtraction

bull Special case of vector addition Add the negative of the subtracted

vectorbull a ndash b = a + (ndash b)

Components of a Vector

bull A component is a part or shadow along a given direction

bull It is useful to use rectangular componentsndash These are the

projections of the vector along the x- and y-axes

Components of a Vector cont

bull The x-component of a vector is the projection along the x-axis

ax = a cosθ

bull The y-component of a vector is the projection along the y-axis

ay = a sinθ

bull a is the magnitude of vector a a2 = ax

2 + ay2

Example 1

Resolve this vector along the x and y axes to find its components respectively

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Geometrical Representation of Vectors

bull Arrows on a plane or spacebull To indicate a vector we use bold letters or an arrow on top of a letter

Properties of Vectors

1 The opposite of a vector a is vector - a

2 It has the same length but opposite direction

3 Two vectors a and b are parallel if one is a positive multiple of the other

a = m b mgt0

Example

if a = 3 b then a is parallel to b

(if a = -2 b then a is anti-parallel to b)

Operations Adding two Vectors

When we add two vectors we get the resultant vector a + b with the parallelogram rule

Operations Adding more vectors

bull We can add more vectors by pairing themappropriately

Operations Vector Subtraction

bull Special case of vector addition Add the negative of the subtracted

vectorbull a ndash b = a + (ndash b)

Components of a Vector

bull A component is a part or shadow along a given direction

bull It is useful to use rectangular componentsndash These are the

projections of the vector along the x- and y-axes

Components of a Vector cont

bull The x-component of a vector is the projection along the x-axis

ax = a cosθ

bull The y-component of a vector is the projection along the y-axis

ay = a sinθ

bull a is the magnitude of vector a a2 = ax

2 + ay2

Example 1

Resolve this vector along the x and y axes to find its components respectively

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Properties of Vectors

1 The opposite of a vector a is vector - a

2 It has the same length but opposite direction

3 Two vectors a and b are parallel if one is a positive multiple of the other

a = m b mgt0

Example

if a = 3 b then a is parallel to b

(if a = -2 b then a is anti-parallel to b)

Operations Adding two Vectors

When we add two vectors we get the resultant vector a + b with the parallelogram rule

Operations Adding more vectors

bull We can add more vectors by pairing themappropriately

Operations Vector Subtraction

bull Special case of vector addition Add the negative of the subtracted

vectorbull a ndash b = a + (ndash b)

Components of a Vector

bull A component is a part or shadow along a given direction

bull It is useful to use rectangular componentsndash These are the

projections of the vector along the x- and y-axes

Components of a Vector cont

bull The x-component of a vector is the projection along the x-axis

ax = a cosθ

bull The y-component of a vector is the projection along the y-axis

ay = a sinθ

bull a is the magnitude of vector a a2 = ax

2 + ay2

Example 1

Resolve this vector along the x and y axes to find its components respectively

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Operations Adding two Vectors

When we add two vectors we get the resultant vector a + b with the parallelogram rule

Operations Adding more vectors

bull We can add more vectors by pairing themappropriately

Operations Vector Subtraction

bull Special case of vector addition Add the negative of the subtracted

vectorbull a ndash b = a + (ndash b)

Components of a Vector

bull A component is a part or shadow along a given direction

bull It is useful to use rectangular componentsndash These are the

projections of the vector along the x- and y-axes

Components of a Vector cont

bull The x-component of a vector is the projection along the x-axis

ax = a cosθ

bull The y-component of a vector is the projection along the y-axis

ay = a sinθ

bull a is the magnitude of vector a a2 = ax

2 + ay2

Example 1

Resolve this vector along the x and y axes to find its components respectively

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Operations Adding more vectors

bull We can add more vectors by pairing themappropriately

Operations Vector Subtraction

bull Special case of vector addition Add the negative of the subtracted

vectorbull a ndash b = a + (ndash b)

Components of a Vector

bull A component is a part or shadow along a given direction

bull It is useful to use rectangular componentsndash These are the

projections of the vector along the x- and y-axes

Components of a Vector cont

bull The x-component of a vector is the projection along the x-axis

ax = a cosθ

bull The y-component of a vector is the projection along the y-axis

ay = a sinθ

bull a is the magnitude of vector a a2 = ax

2 + ay2

Example 1

Resolve this vector along the x and y axes to find its components respectively

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Operations Vector Subtraction

bull Special case of vector addition Add the negative of the subtracted

vectorbull a ndash b = a + (ndash b)

Components of a Vector

bull A component is a part or shadow along a given direction

bull It is useful to use rectangular componentsndash These are the

projections of the vector along the x- and y-axes

Components of a Vector cont

bull The x-component of a vector is the projection along the x-axis

ax = a cosθ

bull The y-component of a vector is the projection along the y-axis

ay = a sinθ

bull a is the magnitude of vector a a2 = ax

2 + ay2

Example 1

Resolve this vector along the x and y axes to find its components respectively

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Components of a Vector

bull A component is a part or shadow along a given direction

bull It is useful to use rectangular componentsndash These are the

projections of the vector along the x- and y-axes

Components of a Vector cont

bull The x-component of a vector is the projection along the x-axis

ax = a cosθ

bull The y-component of a vector is the projection along the y-axis

ay = a sinθ

bull a is the magnitude of vector a a2 = ax

2 + ay2

Example 1

Resolve this vector along the x and y axes to find its components respectively

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Components of a Vector cont

bull The x-component of a vector is the projection along the x-axis

ax = a cosθ

bull The y-component of a vector is the projection along the y-axis

ay = a sinθ

bull a is the magnitude of vector a a2 = ax

2 + ay2

Example 1

Resolve this vector along the x and y axes to find its components respectively

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Example 1

Resolve this vector along the x and y axes to find its components respectively

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Example 2

A vector of 150 N at 120ordm to the x-axis is added to the vector in Example 1 Find the x and y components of the resultant vector

150 N

100 N

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

The Unit Vectors i j k

A unit vector has a magnitude of 1i is the unit vector in the x-direction j is in the y-direction and k is in the z-direction

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

The Unit Vectors Magnitude

Example 3 Given the two displacements

kjid 010306

kjic 080504

Show that the magnitude of e is approximately 17 units where cde 0102

bull Any vector a can be written as

a = x i + y j + z k

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

S - South

N - North

E - EastW - West

15deg

15 deg east of north or 75 deg north of east or

bearing of 15 deg

45 deg west of south or45 deg south of west or

bearing of 225 deg

45deg 30deghellip

hellip

30deg

Direction of Vectors

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Example 4Find the magnitude and direction of the electric field vector E with components 3i ndash 4j

3

4

Note This vector could also be written in matrix form

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

The relative velocity of the cyclist (C) with respect to the pedestrian (P) is given by

VCP = VCE - VPE

Suppose a cyclist (C) travels in a straight line relative to the earth (E) with velocity VCE

A pedestrian (P) is travelling relative to the earth (E) with velocity VPE

Relative velocity

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Example 5

A boat is heading due north as it crosses a wide river with a velocity of 80 kmh relative to water The river has a uniform velocity of 60 kmh due east Determine the velocity (ie speed and direction) of the boat relative to an observer on the riverbank

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

The dot (scalar) product

bull Imagine two vectors a b at an angle θ

bull The dot product is defined to be a b = a b cosθ Useful in finding work of a force F

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

CHECK LISTbull READING

Serwayrsquos Essentials of College Physics pages 41-46 and 53-55Adams and Allday 33 pages 50-51 52-53

Summarybull Be able to give examples of physical quantities

represented by vectors and scalarsbull Understand how to add and subtract vectorsbull Know what a resultant vector isbull Know how to find the change in a vector quantity

calculate relative and resolve a vector into componentsbull Understand how vectors can be represented in

component form in a coordinate systembull Be able to do calculations which demonstrate that you

have understood the above concepts

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples

Numerical Answers for Examplesbull Ex 1 ndash Vx = 87N Vy = 5Nbull Ex 2 ndash coordinates of resultant vector are (116 180)bull Ex 3 ndash length is 169 units or approximately 17 unitsbull Unknown Directions ndash

Yellow is 60deg S of E or 30 deg E of S or bearing 150deg (90+60)Blue is 30deg N of W or 60 deg W of N or bearing 300deg (270+30)

bull Ex 4 ndash Resultant vector E magnitude 5 units ndash Direction 53deg S of E or 37deg E of S or bearing 143deg (90 + 53)

bull Ex 5 ndash velocity of boat relative to earth magnitude 10 kmhrndash Direction 53deg N of E or 37deg E of N or bearing 37deg

• PowerPoint Presentation
• Vectors in Physics-Examples
• Slide 3
• Geometrical Representation of Vectors
• Properties of Vectors
• Operations Adding two Vectors
• Operations Adding more vectors
• Operations Vector Subtraction
• Components of a Vector
• Components of a Vector cont
• Example 1
• Example 2
• Slide 13
• Slide 14
• Slide 15
• Example 4
• Slide 17
• Example 5
• The dot (scalar) product
• CHECK LIST
• Numerical Answers for Examples