Lecture 3(Scalar and Vector Multiplication & 1D

Motion)

Physics 160-02 Spring 2017

Douglas Fields

Multiplication of Vectors

• OK, adding and subtracting vectors seemed fairly straightforward, but how would one multiply vectors?

• There are two ways to multiply vectors and they give different answers…

– Dot (or Scalar) Product

– Cross (or Vector) Product

• You use the two ways for different purposes which will become clearer as you use them.

Dot (or Scalar) Product

• The dot product of two vectors is written as: .

• The result of a dot product is a scalar (no direction).

• There are two ways to find the dot product:

–

– or,

–

• But what IS the dot product – I mean what does it MEAN???

A B

cos cosABA B A B AB

x x y y z zA B A B A B A B

Dot (or Scalar) Product

• The dot product of two vectors gives the length of in the direction of (projection of onto ) times the length of .

• Example:

y

xˆxA i

ˆyA j cosxA A

A

ˆ ˆx yA A i A j

sinyA A

A B

ˆ

ˆ

ˆ ˆ cos 1cos cos

ˆ ˆ cos 1cos 90 sin

Ai

Aj

A i A i A A

A j A j A A

A B BA

B

Dot (or Scalar) Product

• The dot product of two vectors gives the length of in the direction of times the length of .

• Using the other method:

y

xˆxA i

ˆyA j

Aˆˆ ˆ

x y zA A i A j A k

A B

ˆ 1 0 0

ˆ 0 1 0

ˆ 0 0 1

x y z x

x y z y

x y z z

A i A A A A

A j A A A A

A k A A A A

A B B

Dot (or Scalar) Product

• The dot product of two vectors gives the length of in the direction of times the length of .

• Another Example:

y

x

A

A B B

B

cos cos

;

AB B

B A

A B A B A B A B

A B A BA B

B A

A B

Dot (or Scalar) Product

• Physics Example:

• Work – force acting over a distance.

W F D

CPS Question 3-1• Which of the dot products has the greatest absolute

magnitude?A B

A.

B.

C.

D.

y

x

y

x

y

x

y

x

A

A

A

A

B

B

B

B

Dot (or Scalar) Product

• Commutative and Distributive Laws are obeyed by the dot product:

A B B A

A B C A B A C

Dot (or Scalar) Product

• This explains the second method then:

ˆ ˆˆ ˆ ˆ ˆ

ˆˆ ˆ ˆ ˆ ˆ

ˆˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ

x y z x y z

x x x y x z

y x y y y z

z x z y z z

x x y y z z

A B A i A j A k B i B j B k

A i B i A i B j A i B k

A j B i A j B j A j B k

A k B i A k B j A k B k

A B A B A B

Dot (or Scalar) Product

• Usefulness of combining the two methods:

cos

cos

x x y y z z AB

x x y y z z

AB

A B A B A B A B A B

A B A B A B

A B

Cross (or Vector) Product

• The cross product of two vectors is written as: .

• The result of a vector product is a vector (has direction).

• To find the magnitude of a cross product:

–

– Its direction is perpendicular to both and , and given by the Right-Hand-Rule:

A B

sin sinABA B A B AB

A B

Cross (or Vector) Product

• Another way of finding :

–

– Good way to remember using determinant:

A B

ˆˆ ˆy z z y z x x z x y y xA B A B A B i A B A B j A B A B k

ˆˆ ˆ

x y z

x y z

i j k

A B A A A

B B B

Cross (or Vector) Product

• Commutative law is NOT obeyed by the cross product:

–

• Distributive law is obeyed:

–

A B B A

A B C A B A C

Cross (or Vector) Product

• Let’s use this to get the second method:

–

– with

– then

ˆ ˆ ˆˆ ˆ ˆ ˆ ˆ ˆ; ;i j k j k i k i j

ˆ ˆˆ ˆ ˆ ˆ

ˆˆ ˆ ˆ ˆ ˆ

ˆˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆˆ ˆ

x y z x y z

x x x y x z

y x y y y z

z x z y z z

A B A i A j A k B i B j B k

A i B i A i B j A i B k

A j B i A j B j A j B k

A k B i A k B j A k B k

ˆˆ ˆy z z y z x x z x y y xA B A B A B i A B A B j A B A B k

Cross (or Vector) Product

• But what IS the vector product – I mean what does it MEAN???

• It gives a sense of the perpendicularity and length of two vectors.

Cross (or Vector) Product

• Physics Example:

• Torque – what does it take to turn a sticky bolt?

r F

CPS Question 4-1

• ?, , in the x-y planeA B A B

A. 0

B.

C.

D.

E.

y

x

A

B

45o

45o

sin 45oA B

in the positive z-directionA B

in the negative z-directionA B

sin 45 in the negative z-directionoA B

CPS Question 4-2

• What is ?, , in the x-y planeA A B A B

A. 0

B.

C.

D.

E.

y

x

A

B

2

sin 45oA B

2

in the positive z-directionA B

2

in the negative z-directionA B

not enough information

Motion in One Dimension

• We need to define some terms:

– Distance (scalar) [m]

– Displacement (vector) [m]

– Speed (scalar) [m/s]

– Velocity (vector) [m/s]

– Acceleration (vector) [m/s2]

– Time (scalar) [s]

Motion and Graphsy [cm]

x [cm]

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1s 2s 3s 5s4s0s

6s7s8s9s10s

x [cm]

1 2 3 4 5 6 7 8 9t [s]

10

2

4

6

8

10

12

14

16

18

Average Speed and Velocity

• When we talk about speed and velocity, we are referring to changes in distance and displacement over a change in time.

• Important to be specific about these changes:

avg

total distance

total times

avg

total displacement

total time

f i

f i

x xxv

t t t

Average Speed and Velocityx [cm]

1 2 3 4 5 6 7 8 9t [s]

10

2

4

6

8

10

12

14

16

18

• What is the average speed from 0 – 5s? From 0 – 10s?

• What is the average velocity from 0 – 5s? From 0 – 10s?

• Someone walks 100m in what we will call the negative-x direction, then turns around and walks half way back, all at the same pace (speed). It takes 100s to walk the entire path. What is their average velocity?

CPS Question 5-1

A. 1.5m/s

B. 1.5m/s in the +x direction

C. 0.5m/s

D. 0.5m/s in the -x direction

E. 1.0m/s in the -x direction

Instantaneous Velocityx [cm]

1 2 3 4 5 6 7 8 9t [s]

10

2

4

6

8

10

12

14

16

18

• What is the average velocity from 1 – 2s?

• Note that this is the slope of the line connecting these two points.

avg

ˆ ˆ6 2 1ˆ ˆ ˆ4 4 0.042 1 100

f i

f i

x x cm i cm ix cm cm m mv i i i

t t t s s s s cm s

x

t

Instantaneous Velocityx [cm]

1 2 3 4 5 6 7 8 9t [s]

10

2

4

6

8

10

12

14

16

18

• What is the velocity at 2s?

• This is the definition of the derivative of the function that describes the position as a function of time.

• It gives a the slope of the tangent line to the function at any point.

avg0 0

lim limt t

x dxv t v

t dt

Instantaneous Velocityv [cm/s]

1 2 3 4 5 6 7 8 9t [s]

10

2

4

6

• We can now plot the velocity as a function of time.

• Notice that the velocity also changes.

• So, we can ask, what is the change in velocity as a function of time?

• This is known as the acceleration.

-6

-4

-2

Exercise 2.10

Exercise 2.11

Instantaneous Accelerationv [cm/s]

1 2 3 4 5 6 7 8 9 10

2

4

6

• The acceleration is just given by the derivative of the velocity function.

• So, it is also the second derivative of the position function.

-6

-4

-2

2

avg 20 0lim limt t

v dv d dx d xa t a

t dt dt dt dt

t [s]

Instantaneous Acceleration

a [cm/s2]

1 2 3 4 5 6 7 8 9 t [s]10

2

4

6

-6

-4

-2

v [cm/s]

1 2 3 4 5 6 7 8 9t [s]

10

2

4

6

-6

-4

-2

Velocity and acceleration

Exercise 2.12

Curvaturex [cm]

1 2 3 4 5 6 7 8 9

t

10

2

4

6

8

10

12

14

16

18

a [cm/s2]

1 2 3 4 5 6 7 8 9 t [s]10

2

4

6

-6

-4

-2

Spherical Cows

• http://en.wikipedia.org/wiki/Spherical_cow• Milk production at a dairy farm was low, so the farmer wrote to the local

university, asking for help from academia. A multidisciplinary team of professors was assembled, headed by a theoretical physicist, and two weeks of intensive on-site investigation took place. The scholars then returned to the university, notebooks crammed with data, where the task of writing the report was left to the team leader. Shortly thereafter the physicist returned to the farm, saying to the farmer "I have the solution, but it only works in the case of spherical cows in a vacuum."