By:

Engr. Hinesh KumarLecturer

I.B.T, LUMHS, Jamshoro

Scalars are quantities which have magnitude without direction.

Examples of scalars

• temperature• mass• kinetic energy

• time• amount• density• charge

Scalars

VectorA vector is a quantity that has both magnitude (size) and direction.

it is represented by an arrow whereby– the length of the arrow is the magnitude, and– the arrow itself indicates the direction

Contd….The symbol for a vector is a letter with an

arrow over it.

All vectors have head and tail.

A

Two ways to specify a vector

It is either given by• a magnitude A, and• a direction

Or it is given in the x and y components as

• Ax

• Ay

y

x

A

A

Ay

x

Ax

Ay

y

x

AAx

AyA

Ax = A cos

Ay = A sin

│A │ =√ ( Ax2

+ Ay2

)

The magnitude (length) of A is found by using the Pythagorean Theorem

The length of a vector clearly does not depend on its direction.

y

x

AAx

AyA

The direction of A can be stated as

tan = Ay / Ax

=tan-1(Ay / Ax)

Vector Representation of ForceForce has both magnitude and direction and

therefore can be represented as a vector.

Vector Representation of Force

The figure on the left shows 2 forces in the same direction therefore the forces add. The figure on the right shows the man pulling in the opposite direction as the cart and forces are subtracted.

Some Properties of Vectors

Equality of Two Vectors

Two vectors A and B may be defined to be equal if they have the same magnitude and point in the same directions. i.e. A = B

A BA

A

B

B

Negative of a VectorThe negative of vector A is defined as giving the vector sum of zero value when added to A . That is, A + (- A) = 0. The vector A and –A have the same magnitude but are in opposite directions.

A

-A

Applications of VectorsVECTOR ADDITION – If 2 similar vectors point in

the SAME direction, add them.

Example: A man walks 54.5 meters east, then another 30 meters east. Calculate his displacement relative to where he started?

54.5 m, E 30 m, E+

84.5 m, E

Notice that the SIZE of the arrow conveys MAGNITUDE and the way it was drawn conveys DIRECTION.

The addition of two vectors A and B

- will result in a third vector C called the resultant

C = A + B

A

BC

Geometrically (triangle method of addition)

• put the tail-end of B at the top-end of A• C connects the tail-end of A to the top-end of B

We can arrange the vectors as we like, as long as we maintain their length and direction

Vector Addition

ExampleExample

x1

x5

x4

x3

x2

xi

xi = x1 + x2 + x3 + x4 + x5

ExampleExample

Applications of VectorsVECTOR SUBTRACTION - If 2 vectors are going

in opposite directions, you SUBTRACT.

Example: A man walks 54.5 meters east, then 30 meters west. Calculate his displacement relative to where he started?

54.5 m, E

30 m, W-

24.5 m, E

Vector SubtractionEquivalent to adding the negative vector

A

-BA - B

B

A BC =

A + (-B)C =

ExampleExample

Scalar MultiplicationThe multiplication of a vector Aby a scalar

- will result in a vector B

B = A- whereby the magnitude is changed but not the direction

• Do flip the direction if is negative

B = A

If = 0, therefore B = A = 0, which is also known as a zero vector

(A) = A = (A)

(+)A = A + A

ExampleExample

Rules of Vector Addition commutative

A + B = B + A

A

B

A + BB

A A + B

associative

(A + B) + C = A + (B + C)

B

CA

B

CA A + B

(A + B) + CA + (B + C)

B + C

distributive

m(A + B) = mA + mB

A

B

A + B mA

mB

m(A + B)

Parallelogram method of addition (tailtotail)

A

B

A + B

The magnitude of the resultant depends on the relative directions of the vectors

a vector whose magnitude is 1 and dimensionless

the magnitude of each unit vector equals a unity; that is, │ │= │ │= │ │= 1

i a unit vector pointing in the x direction

j a unit vector pointing in the y direction

k a unit vector pointing in the z direction

and defined as

Unit Vectors

k

j

i

Useful examples for the Cartesian unit vectors [ i, j, ki, j, k ] - they point in the direction of the x, y and z axes respectively

x

y

z

ii

jj

kk

Component of a Vector in 2-D vector A can be resolved into two

components Ax and Ay

x- axis

y- axis

Ay

Ax

A

θ

A = Ax + Ay

The component of A are

│Ax│ = Ax = A cos θ

│Ay│ = Ay = A sin θ

The magnitude of A

A = √Ax2 + Ay

2

tan = Ay / Ax

=tan-1(Ay / Ax)

The direction of A

x- axis

y- axis

Ay

Ax

A

θ

The unit vector notation for the vector AA is written

A = Axi + Ayj

x- axis

y- axis

Ax

Ay

θ

A

i

j

Component of a Unit Vector in 3-D vector A can be resolved into three

components Ax , Ay and Az

A

Ax

Ay

Az

y- axis

x- axis

z- axis

i

j

k

A = Axi + Ayj + Azk

if

A = Axi + Ayj + Azk

B = Bxi + Byj + Bzk

A + B = C sum of the vectors A and B can then be obtained as vector C

C = (Axi + Ayj + Azk) + (Bxi + Byj + Bzk)

C = (Ax + Bx)i+ (Ay + By)j + (Az + Bz)kC = Cxi + Cyj + Czk

Dot product (scalar) of two vectors

The definition:

θ

B

AA · B = │A││B │cos θ

if θ = 900 (normal vectors) then the dot product is zero

Dot product (scalar product) properties:

if θ = 00 (parallel vectors) it gets its maximum

value of 1

and i · j = j · k = i · k = 0|A · B| = AB cos 90 = 0

|A · B| = AB cos 0 = 1 and i · j = j · k = i · k = 1

A + B = B + A

the dot product is commutative

Use the distributive law to evaluate the dot product

if the components are known

A · B = (Axi + Ayj + Azk) · (Bxi + Byj + Bzk)

A. B = (AxBx) i.i + (AyBy) j.j + (AzBz) k.k

A .

Cross product (vector) of two vectorsThe magnitude of the cross product given by

the vector product creates a new vector

this vector is normal to the plane defined by the

original vectors and its direction is found by using the

right hand rule

│C │= │A x B│ = │A││B │sin θ

θ

A

BC

if θ = 00 (parallel vectors) then the cross

product is zero

Cross product (vector product) properties:

if θ = 900 (normal vectors) it gets its maximum

value

and i x i = j x j = k x k = 0|A x B| = AB sin 0 = 0

|A x B| = AB sin 90 = 1 and i x i = j x j = k x k = 1

the relationship between vectors i , j and k can

be described as

i x j = - j x i = k

j x k = - k x j = i

k x i = - i x k = j

Example 1 (2 Dimension)

If the magnitude of vector A and B are equal to 2 cm and 3 cm respectively , determine the magnitude and direction of the resultant vector, C for

B

Aa) A + B

b) 2A + B

Solution

a) |A + B| = √A2 + B2

= √22 + 32

= 3.6 cm

The vector direction

tan θ = B / A

θ = 56.3

b) |2A + B| = √(2A)2 + B2

= √42 + 32

= 5.0 cm

The vector direction

tan θ = B / 2A

θ = 36.9

Example 2

Find the sum of two vectors A and B lying in the xy plane and given by

A = 2.0i + 2.0j and B = 2.0i – 4.0j

SolutionComparing the above expression for A with the general relation A = Axi + Ayj , we see that Ax= 2.0 and Ay= 2.0. Likewise, Bx= 2.0, and By= -4.0 Therefore, the resultant vector C is obtained by using EquationC = A + B + (2.0 + 2.0)i + (2.0 - 4.0)j = 4.0i -2.0j

or Cx = 4.0 Cy = -2.0

The magnitude of C given by equation

C = √Cx2 + Cy

2 = √20 = 4.5

Find the angle θ that C makes with the positive x axis

Exercise

(1, 0)

(2, 2)

x1 + x2 = (1, 0) + (2, 2)= (3, 2)

x1

x2

x1 + x2

(1, 0)

(2, 2)

(x2)

x1

x1 + x2x2

x1 + x2 = (1, 0) + (2, 2)= (3, 2)

x1 - x2?

(1, 0)

(2, 2)

x1

-x2x1 - x2

x1 - x2 = (1, 0) - (2, 2)= (-1, -2)

x1 - x2 = x1 + (-x2)

Example -2D for subtraction

(1, 0)

(2, 2)

AssignmentIf one component of a vector is not zero, can its magnitude be zero? Explain and Prove it.

If A + B = 0, what can you say about the components of the two vectors?

A particle undergoes three consecutive displacements d1 = (1.5i + 3.0j – 1.2k) cm,

d2 = (2.3i – 1.4j – 3.6k) cm d3 = (-1.3i + 1.5j) cm. Find the component and its magnitude.

1

2

3