Chapter 3Chapter 3

Vectors and Vectors and

Two-Dimensional MotionTwo-Dimensional Motion

Vectors and ScalarsVectors and Scalars

A A scalarscalar is a quantity that is is a quantity that is completely specified by a positive or completely specified by a positive or negative number with an appropriate negative number with an appropriate unit and unit and has no directionhas no direction..

A A vectorvector is a physical quantity that is a physical quantity that must be described by a magnitude must be described by a magnitude (number) and appropriate units (number) and appropriate units plus plus a directiona direction..

Scalar Quantity Scalar Quantity a quantity that has a quantity that has magnitudemagnitude but but

not directionnot direction

Vector Quantity Vector Quantity a quantity that has a quantity that has both both

magnitude and directionmagnitude and direction

Magnitude – the numerical value of Magnitude – the numerical value of a scalar or vector.a scalar or vector.

For example, a velocity vector might For example, a velocity vector might be 30 m/s to the west. The be 30 m/s to the west. The magnitude of this vector is 30 m/s. magnitude of this vector is 30 m/s. A force vector might be 100 A force vector might be 100 pounds upward. The magnitude of pounds upward. The magnitude of this vector is 100 pounds.this vector is 100 pounds.

Scalar QuantitiesScalar Quantitieslengthlength

e.g. 93,000,000 milese.g. 93,000,000 milesmassmass

e.g. 180 kge.g. 180 kgspeedspeed

e.g. 186,000 miles/seconde.g. 186,000 miles/second

More examplesMore examples Temperature (20Temperature (20o o C)C) Volume (45 cmVolume (45 cm3)3)

Time intervals (24 h)Time intervals (24 h)

Rules of ordinary arithmetic are Rules of ordinary arithmetic are used to manipulate scalar used to manipulate scalar quantitiesquantities

Vector QuantitiesVector Quantities forceforce

e.g. 20 Newtons Eastwarde.g. 20 Newtons Eastwardvelocityvelocity

e.g. 20 meters/second e.g. 20 meters/second NorthNorth

accelerationacceleratione.g. 9.8 m/se.g. 9.8 m/s22 downward downward

VectorsVectors

are used to denote quantities that are used to denote quantities that have have magnitudemagnitude and and directiondirection

can be added and subtractedcan be added and subtracted can be multiplied or divided by a can be multiplied or divided by a

numbernumber can be manipulated can be manipulated graphicallygraphically ( (i.e.,i.e.,

by drawing them out) or algebraically by drawing them out) or algebraically (by considering components)(by considering components)

VectorVectoran arrow drawn to scale is an arrow drawn to scale is used to represent a vector used to represent a vector quantityquantity

vector notationvector notation

F

Vector NotationVector Notation

When handwritten, use an arrow:When handwritten, use an arrow:

When printed, will be in bold print: When printed, will be in bold print: AA

A

Vector ExampleVector Example A particle travels from A A particle travels from A

to B along the path to B along the path shown by the dotted red shown by the dotted red lineline This is the This is the distance distance

traveled and is a scalartraveled and is a scalar The The displacementdisplacement is is

the solid line from A to Bthe solid line from A to B The displacement is The displacement is

independent of the path independent of the path taken between the two taken between the two pointspoints

Displacement is a vectorDisplacement is a vector Notice the arrow Notice the arrow

indicating directionindicating direction

Properties of VectorsProperties of Vectors

Equality of Two VectorsEquality of Two Vectors Two vectors are Two vectors are equalequal if they have if they have

the same magnitude the same magnitude andand the same the same directiondirection

Movement of vectors in a diagramMovement of vectors in a diagram Any vector can be moved parallel to Any vector can be moved parallel to

itself without being affecteditself without being affected

Equivalent Equivalent VectorsVectors

Vectors are said to be equivalent if both Vectors are said to be equivalent if both their magnitudes and direction matchtheir magnitudes and direction match

Adding VectorsAdding Vectors

When adding vectors, their When adding vectors, their directions must be taken into directions must be taken into accountaccount

Units must be the same Units must be the same Graphical MethodsGraphical Methods

Use scale drawingsUse scale drawings Accuracy difficult to controlAccuracy difficult to control

Algebraic MethodsAlgebraic Methods Accuracy well definedAccuracy well defined

More Properties of VectorsMore Properties of Vectors

Resultant VectorResultant Vector The The resultantresultant vector is the sum of a vector is the sum of a

given set of vectorsgiven set of vectors R A B

Graphical Methods of Vector Graphical Methods of Vector AdditionAddition

tip-to-tail methodtip-to-tail method

parallelogram methodparallelogram method

The Tip-to-Tail Method

To add vector To add vector BB to vector to vector AA::

Draw vector Draw vector AA..

Draw vector Draw vector BB with its tail with its tail starting from the tip of starting from the tip of AA..

The sum vector The sum vector AA++BB is the is the vector drawn from the tail vector drawn from the tail of of vector vector AA to the tip of vector to the tip of vector BB..

AB

C

DE

F

(A = 4 cm; B = 2 cm; C = 3 cm; D = 2 cm; E = 3 cm; F = 2 cm)

A + B = ?B

AA + B

AD

A + D = ?

A + D

C

D

C + D = ?

C + D

B

E

B + E = ?

B + E

A

F

A + F = ?

A + F

Adding Vectors Adding Vectors Graphically, cont.Graphically, cont.

When you have When you have many vectors, just many vectors, just keep repeating the keep repeating the process until all process until all are includedare included

The The resultantresultant is is still drawn from still drawn from the origin of the the origin of the first vector to the first vector to the end of the last end of the last vectorvector

A

B

D

A + B + D = ?

A + B + D

The order in which the vectors are added does not matter.

A

B

D

D + B + A = ?

D + B + A

A

B

D

A + B + D

A

B

C

D

F

A + B + D + C + F = ?

A + B + D + C + F

Why we need to use Why we need to use vectors?vectors?

Riverboat:Riverboat:

V

VC

V=Velocity of boat in calm water.VC = Velocity of CurrentVR = Resulting Velocity

VR

V

Vector vs. Scalar ReviewVector vs. Scalar Review

A A vectorvector quantity has both quantity has both magnitude (size) and directionmagnitude (size) and direction

A A scalarscalar is completely specified by is completely specified by only a magnitude (size)only a magnitude (size)

Alternative Graphical Alternative Graphical MethodMethod

When you have only When you have only two vectors, you may two vectors, you may use the use the Parallelogram Parallelogram MethodMethod

All vectors, including All vectors, including the resultant, are the resultant, are drawn from a common drawn from a common originorigin The remaining sides of The remaining sides of

the parallelogram are the parallelogram are sketched to determine sketched to determine the diagonal, the diagonal, RR

A + B = ?

B

A

A + B

Multiplying or Dividing a Multiplying or Dividing a Vector by a ScalarVector by a Scalar

The result of the multiplication or division The result of the multiplication or division is a vectoris a vector

The magnitude of the vector is multiplied The magnitude of the vector is multiplied or divided by the scalaror divided by the scalar

If the scalar is positive, the direction of the If the scalar is positive, the direction of the result is the same as of the original vectorresult is the same as of the original vector

If the scalar is negative, the direction of If the scalar is negative, the direction of the result is opposite that of the original the result is opposite that of the original vectorvector

A

BB

A

A + B =

B - A =

A + B =

A - B =

A

B

A

B

A + B =

B - A =

A + B =

A - B =

Find the vector Find the vector “B – A”“B – A”

1.1.

2.2.

3.3.

4.4.

A

B

Two unknown vectors Two unknown vectors AA and and BB are added. are added.

The magnitude of the sum vector The magnitude of the sum vector “A + “A + B” B” (i.e., the quantity (i.e., the quantity |A + B||A + B|))1.1. is at least as great as is at least as great as |A||A| ( (i.e.,i.e., the the

magnitude of magnitude of AA). ).

2.2. is at most as great as is at most as great as |A||A| + + |B||B| ( (i.e.,i.e., the magnitudes of the magnitudes of AA and and BB added). added).

3.3. must be equal to must be equal to |A||A| + + |B||B|. .

4.4. can be greater than can be greater than |A||A| + + |B||B|. .