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AP Physics Chapter 3

Vector

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AP Physics

Turn in Chapter 2 Homework, Worksheet, & Lab

Take quiz Lecture Q&A

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Vector and Scalar

Vector: – Magnitude: How large, how fast, … – Direction: In what direction (moving or pointing)– Representation depends on frame of reference

Scalar: – Magnitude only– No direction– Representation does not depend on frame of

reference

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Examples of vector and scalar

Vectors:– Position, displacement, velocity, acceleration, force,

momentum, …

Scalars:– Mass, temperature, distance, speed, energy,

charge, …

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Vector symbol

Vector: bold or an arrow on top and v V

– Typed: v and V or

Scalar: regular– v or V

v stands for the magnitude of vector v.

– Handwritten: and v V

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Adding Vectors

Graphical– Head-to-Tail (Triangular)– Parallelogram

Analytical (by components)

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Graphical representation of vector: Arrow

An arrow is used to graphically represent a vector.

ab

– The direction of the arrow represents the direction of the vector.– When comparing the magnitudes of vectors, we ignore

directions.

head

tail

– The length of the arrow represents the magnitude of the vector.

Vector a is smaller than vector b because a is shorter than b.

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Equivalent Vectors

Two vectors are identical and equivalent if they both have the same magnitude and are in the same direction.

AB C

– A, B and C are all equivalent vectors.

– They do not have to start from the same point. (Their tails don’t have to be at the same point.)

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Negative of Vector

Vector -A has the same magnitude as vector A but points in the opposite direction.

If vector A and B have the same magnitude but point in opposite directions, then A = -B, and B = -A

A -A

-A

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Adding Vectors: Head-to-Tail

Head-to-Tail method:A

B

A

BA+B

Make sure arrows are parallel and of same length.

– Draw vector A– Draw vector B starting

from the head of A– The vector drawn from the

tail of A to the head of B is the sum of A + B.

Example: A + B

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A+B=B+A

A

A

A

B

B

BA+BB+A

A + B

How about B + A?

What can we conclude?

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A+B+C

A

B

A+B+C

C

A B C

Resultant vector:

from tail of first to head of last.

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A-B=A+(-B)

A B

-B

A

A-B

– Draw vector A– Draw vector -B from head

of A.– The vector drawn from the

tail of A to head of –B is then A – B.

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What Are the Relationships?

cb

a a

bc

b c a

0a b c

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Magnitude of sum A + B = C

|A – B| C A + B

A and B in same direction

A and B in opposite direction

A and B at some angle

max. c min. c

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Adding Vectors: Parallelogram

A+B

A+B

A

B

– Draw the two vectors from the same point

Advantage:No need to measure length.

– Construct a parallelogram with these two vectors as two adjacent sides

– The sum is the diagonal vector starting from the same tail point.

A B

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Example

Vector a has a magnitude of 5.0 units and is directed east. Vector b is directed 35o west of north and has a magnitude of 4.0 units. Construct vector diagrams for calculating a + b and b – a. Estimate the magnitudes and directions of a + b and b – a from your diagram.

E

N

W

S

35o

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Solution

35o

a

ba+

b

-a

b-a

a+b: 4.3 unit, 50o North of East

b-a: 8.0 unit, 66o West of North

E

S

W

N

Using ruler and protractor, we find:

50o 66o

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Vector Components

Drop perpendicular lines from the head of vector a to the coordinate axes, the components of vector a can be found:

cos

sinx

y

a a

a a

x

y

ax

aya

is the angle between the vector and the +x axis.

ax and ay are scalars.

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Finding components of a vector

Resolving the vector Decomposing the vector

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Vector magnitude and direction

The magnitude and direction of a vector can be found if the components (ax and ay) are given:

2 2magnitude: x ya a a

is the angle from the +x axis to the vector.

(for 3-D)2 2 2x y za a a

1 directi : non ta y

x

a

a

a

ax

ay

x

y

tan y

x

a

a

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Example

A ship sets out to sail to a point 120 km due north. Before the voyage, an unexpected storm blows the ship to a point 100 km due east of its starting point. How far, and in what direction, must it now sail to reach its original destination?

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Solution

It must sail 156 km at 39.8o West of North to reach its original destination.

a =

120

b = 100

c

E

Nc

2 2 2 2120 100 156a b km

1 1 100tan tan 39.8

120ob

a

A ship sets out to sail to a point 120 km due north. Before the voyage, an unexpected storm blows the ship to a point 100 km due east of its starting point. How far, and in what direction, must it now sail to reach its original destination?

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Practice: What are the magnitude and direction of vector ˆ ˆ4 5 ?a i j

x

y

5

4a 2 2 2 24 5 6.4x ya a

1 1 5tan tan 51

4y o

x

a

a

4, 5,

?, ?

x ya a

a

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Practice: What are the components of a vector that has a magnitude of 12 units and makes an angle of 126o with the positive x direction?

x

y

126o

12, 126

?, ?

o

x y

a

a a

cos 12cos126 7.1oxa a

sin 12sin126 9.7oya a

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Unit vectors

Unit vector: magnitude of exactly 1 and points in a particular direction.– x direction:– y direction:

– z direction:

ii or

jj or

kk or

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Vector components and expression

Any vector can be written in its components and the unit vectors:

ˆˆ

ˆˆ ˆ

ˆ

x y

y z

z

xa i a j a ka

b b i b j b k

, ,x y za a a

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Terminology

axi is the vector component of a.

ax is the (scalar) component of a.

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Example

Express the following vector in component and unit vector form.

x

y

=30o

a=12.0 units

xa

ˆ ˆ ˆ ˆ10.4 6.00x ya a i a j i j ya

cos 12.0cos30 10.4oa

sin 12.0sin30 6.00oa

ay

ax

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Adding Vectors by Components

When adding vectors by components, we add components in a direction separately from other components.

r a b

r a b

s a b

3-D

x

y

ax

ay bx

by

a

b

r

y y yr a b xr x xa b

Component form:

rx

ry

z z zr a b

2-D

ˆˆ ˆx x y y z za b i a b j a b k

ˆˆ ˆx x y y z za b i a b j a b k

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Example

The minute hand of a wall clock measures 10 cm from axis to tip. What is the displacement vector of its tip (a) from a quarter after the hour to half past, (b) in the next half hour, and (c) in the next hour?

•

•

•

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Solution

O A (10, 0)

C (0,10)

B (0,-10)

)

A B

a

r

rAB

rBC

x

y

)

(0,10) (0, 10)

(0,20)

B C C B

b

r r r

cm

)

0C C

c

r r r

B Ar r(0, ) (11 ,0)0 0

(0 10, ) ( 10, 1010 0 )cm

or magnitude and direction form.

Try b) and c)

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Practice

)

ˆ ˆˆ ˆ ˆ3 4ˆ4

a

a b i j k i j k

ˆ ˆˆ ˆ ˆ ˆ4

ˆ ˆˆ ˆ ˆ ˆ(4 1)

3

3 1( ) ( ) 5 44 3

4

1

a b i j k i j k

i j k i j k

)

0

c

a b c

ˆ ˆˆ ˆ ˆ ˆ(4 1) ( ) ( )43 3 21 1 5i j k i j k

c ˆˆ ˆ( ) 5 4 3a b i j k

Two vectors are given by a = 4i – 3j + k and b = -i + j + 4k. Find:

a) a + b

b) a – b

c) a vector c such that a – b + c = 0

)b

b a

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Practice: 54-19The two vectors a and b in Fig. 3-29 have equal magnitudes of 10.0 m and the angels are 1 = 30o and 2 = 105o. Find the (a) x and (b) y components of their vector sum r, (c) the magnitude of r, and (d) the angle r makes with the positive direction of the x axis.

) )

cos 10cos30 8.66

sin 10sin30 5

ox

oy

a b

a a

a a

y

x

b

a

rb=105o+30o=135o

a

07.7135sin10

07.7135cos10o

y

ox

b

b

x

y

r

r

1

2

sin

cos

aa

aa

y

x

8.66 7.07 1.59x xa b

5 7.07 12.07y ya b

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54-19 (Continued)

)c

r

)

r

d

2 2 2 21.59 12.07 12.2x yr r

1 1 12.07tan tan 82.5

1.59y o

x

r

r

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Vector Multiplication

More:

ˆˆ ˆ2 4a i j k

ˆ ˆˆ ˆ ˆ ˆ4 4 2 4 4 4 1 8 16 4a i j k i j k

Scalar (aka dot or inner) product: a b Vector (aka cross) product: a b

We cannot write ab if a and b are vectors. But we still can write 2a since 2 is a scalar.

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Scalar product: ab

(phi) is the angle between vector a and b. is always between 0o and 180o. (0o 180o) The scalar product is a scalar It has no

direction. What if the two vectors are perpendicular to each

other?

cosa b ab a

b

90o 0a b

: angle between vector and +x axis

: angle between two vectors

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Physical meaning of ab

ba is the projection of b onto a.

ba

ab

a

b

cosab b

cosba a a

b

ab is the project of a onto b.

Also

cos aba b a ab

cos bba b a a b

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Properties of scalar product

1. ab = ba2. ii = jj = kk = 13. ij = ji = jk = kj = ki = ik = 04. aa = a2

5. ab = 0 if a b

ˆ ˆˆ ˆ ˆ ˆx y z x y za b a i a j a k b i b j b k

6. ab = axbx+ayby+azbz

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Angle between two vectors

cosa b ab

When we know the components:

When we know magnitudes and :

x x y y z za b a b a b a b

cos x x y y z zab a b a b a b Put together:

cos x x y y z za b a b a b

ab

1cos x x y y z za b a b a b

ab

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Example:a. Determine the components and magnitude of r = a – b + c if a = 5.0i + 4.0j – 6.0k, b = -2.0i + 2.0j + 3.0k, and c = 4.0i + 3.0 j + 2.0k.b. Calculate the angle between r and the positive z axis.

)

ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ(5.0 ) ( 2.0 ) (4.0 )

ˆˆ ˆ(5.0

6.0 3.0 2.0

6

4

2.0 4.0) ( ) ( )

ˆˆ ˆ11.0 5

.0 2.0 3

.0 3.0

.0

4.0 2 2.0

.0 7.0

.0 3.0

a

r a b c

i j k i j k i j k

i j k

i j k

x

z

O

B

A

2 2

)

11.0 5.0 12.1

b

OA y

11

5

-7

7.0AB

90oOAB

tan AOB

1tan 0.579 30.0oAOB

90 120.0o oCOB AOB

7.00.579

12.1

AB

OA

C•

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Another approach

ˆˆ ˆ11.0 5.0 7.0r i j k

We are looking for the angle between r and any vector in the z direction. Let’s choose the unit vector in the z direction, k

ˆ ˆ0ˆ ˆ, a 0nd 1k i j k

ˆr k 11 0 5 0 7 1 7x x y y z zr k r k r k

22 211 5 7 13.9r

1k

ˆ cosr k rk

ˆ 7cos 0.5

13.9 1

r k

rk

1cos 0.5 120o

cosa b ab

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Practice: 55-43For the vectors in Fig. 3-35, with a = 4, b =3, and c =5, calculate

b

a

c5 3

4

)

0

a

a b

(a) , (b) , and (c) .a b a c b c

a b

1 1

)

3tan tan 36.87

4o

b

b

a

180 36.87 143.17o o o

cos 4 5cos143.17 16oa c ac

A

B

1 1

)

4tan tan 53.13

3o

b

aA

b 180 53.13 127o o oB

cos 3 5cos127 9ob c bc B

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Approach 2

x

y

a

c bb

c

ˆ ˆ4 0a i j

ˆ0 3b i j

ˆ ˆ4 3c i j

a c

b c

-4

-3 4 4 0 3 16

0 4 3 3 9

a b

4 0 0 3 0

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Approach 3

b

a

c5 3

4

)aa b

)b

c a b

a c a a b a a a b

2 16a )c

0a b c

b c b a b a b

2 9b b b

0a b

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Approach 4

b

a

c5 3

4

)b

a c

)c

b c

a c 16

da

d = -c

ce

aad aa

b c

b b 9eec

e = -b

a d ��������������

e c

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Vector Product: c = a b

Magnitude of c is:

– Place the vectors a and b so that their tails are at the same point.

– Extend your right arm and fingers in the direction of a.– Rotate your hands along your arm so that you can flap

your fingers toward b through the smaller angle between a and b. Then

– Your outstretched thumb points in the direction of c.

A

A B

c is a vector, and it has a direction given by the right-hand-rule (RHR):

sinc ab

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Properties of cross product

b a = - (a b) a b is a, and a b is ba a = 0

jik

ikj

kji

ˆˆˆ

ˆˆˆ

ˆˆˆ

kbabajbabaibababa xyyxzxxzyzzyˆˆˆ

ˆ ˆˆ ˆ ˆ ˆ

x y z x y za b a i a j a k b i b j b k

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Practice:Three vectors are given by a = 3.0i + 3.0j – 2.0k, b = -1.0i –4.0j + 2.0k, and c = 2.0i + 2.0j + 1.0k. Find (a) a • (b c), (b) a • (b + c), and (c) a (b + c).

)a

b c

( )a b c

ˆ( 4.0)(1.0) (2.0)(2.0)

ˆ (2.0)(2.0) ( 1.0)(1.0)

ˆ ˆˆ ˆ ( 1.0)(2.0) ( 4.0)(2.0) 8.0 5.0 6.0

i

j

k i j k

ˆ ˆˆ ˆ ˆ ˆ(3.0 3.0 2.0 ) ( 8.0 5.0 6.0 )i j k i j k

(3.0)( 8.0) (3.0)(5.0) ( 2.0)(6.0) 21

ˆ ˆˆ ˆ ˆ ˆ1.0 4.0 2.0 2.0 2.0 1.0i j k i j k

kbabajbabaibababa xyyxzxxzyzzyˆˆˆ

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Pg52-53P (2)

)

ˆ ˆˆ ˆ ˆ ˆ1.0 4.0 2.0 2.0 2.0 1.0

b

b c i j k i j k

( )a b c

ˆˆ ˆ( 1.0 2.0) ( 4.0 2.0) (2.0 1.0)

ˆˆ ˆ1.0 2.0 3.0

i j k

i j k

ˆ ˆˆ ˆ ˆ ˆ(3.0 3.0 2.0 ) (1.0 2.0 3.0 )i j k i j k

(3.0)(1.0) (3.0)( 2.0) ( 2.0)(3.0) 9

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Pg52-53P (3)

kjicb

c

ˆ0.3ˆ0.2ˆ0.1

)

( )a b c ˆ ˆˆ ˆ ˆ ˆ(3.0 3.0 2.0 ) (1.0 2.0 3.0 )i j k i j k

ˆ(3.0)(3.0) ( 2.0)( 2.0)

ˆ ( 2.0)(1.0) (3.0)(3.0)

ˆ (3.0)( 2.0) (3.0)(1.0)

ˆˆ ˆ5.0 11.0 9.0

i

j

k

i j k

52

Law of cosine

Cabbac cos2222

b

a

c

C

53

Law of Sine

C

c

B

b

A

a

sinsinsin

a

c

b

B A

C

sin sin sinA B C

a b c or