Fun with Vectors

Definition

• A vector is a quantity that has both magnitude and direction

• Examples?

Represented by an arrow

A (initial point)

B (terminal point)

v, v, or AB

If two vectors, u and v, have the same length and direction, we say they are equivalent

uv

ab

Vector addition

ab

Vector addition: a + b

a

b

Vector addition: a + b

a+b

a

b

Vector addition: a + b

a+b

Scalar Multiplication

a

Scalar Multiplication

a2a

-a -½ a

Subtraction

ba

Subtraction

ba

-b

Subtraction

ba

-ba+(-b)

Subtraction

ba

-ba+(-b)

Subtraction

ba

-b

a+(-b)

If a and b share the same initial point, the vector a-b is the vectorfrom the terminal point of b to the terminal point of a

Let’s put these on a coordinate system

We can describe a vector by putting its initial point at the origin.

We denote this as a=<a1,a2>

where (a1,a2) represent the terminal point

Graphically

x

y

a=<a1,a2>

(a1,a2)

y

x

z

v=<a,b,c>

a

b

c

(a,b,c)

Given two points A=(x1,y1) and B=(x2,y2),

The vector v = AB is given by

v = <x2 - x1, y2 - y1>

…or in 3-space, v = <x2 - x1, y2 - y1, z2 - z1>

Graphically

A=(-1,2)B=(2,3)

A

B

v = <2-(-1), 3-2> = <3,1>

v

Recall, a vector has direction and length

Definition: The magnitude of a vector v = <x,y,z> is given by

222 zyxv

Properties of VectorsSuppose a, band c are vectors, c and d are scalars

1. a+b=b+a2. a+(b+c)=(a+b)+c3. a+0=a4. a+(-a)=05. c(a+b)=ca+cb6. (c+d)a=ca+da7. (cd)a=c(da)8. 1a=a

Standard Basis Vectors

1

1,0,0

0,1,0

0,0,1

kji

k

j

i

Definition: vectors with length 1 are called unit vectors

Example: We can express vectors in terms of this basis

a = <2,-4,6> a = 2i -4j+6k

Q. How do we find a unit vector in the same direction as a?

A. Scale a by its magnitude

11

1

a

aa

av

aa

v

6,4,256

1

6,4,26)4(2

1

1

222

aa

v

Examplea = <2,-4,6>

12.3 The Dot Product

Motivation: Work = Force* Distance

Box

F

D

Fx

Fy

Box

DF

Fx

Fy

To find the work done in moving the box, we want the part of F in the direction of the distance

)cos(FFx

)cos( FDW

One interpretation of the dot product

)cos( DFDF

Where is the angle between F and D

A more useful definition

332211 babababa

)cos( 2222

bababa

You can show these two definitions are equal by considering the following triangle and applying the law of cosines! See page 808 for details

y

x

z

ba-b

aThink, what is |a|2?

Example

a=<2,-1,0>, b=<1,-8,-3>

Find a.b and the angle between a and b

The Dot Product

If a = <a1,a2,a3> and b=<b1,b2,b3> then

The dot product of a and b is a NUMB3R given by

332211 bababa ba

)cos(baba

The Dot Product

baba

)cos( 0)2/cos(

0)cos(2

0

a and b are orthogonal if and only if the dot product of a and b is 0

Other Remarks:

a

b

0)cos(2

Properties of the dot productSuppose a, b, and c are vectors and c is a scalar

1. a.a=|a|2

2. a.b=b.a

3. a.(b+c) = (a.b)+(a.c)

4. (ca).b=c(a.b)=a.(cb)

5. 0.a=0

Yet another use of the dot product: Projections

a.b=|a| |b| cos( )

Think of our work example: this is ‘how much’ of b is in the direction of a

b

a|b| cos( )

We call this quantity the scalar projection of b on a

)cos(ba

babcompa

Think of it this way: The scalar projection is the length of the shadow of b cast upon a by a light directly above a

Q. How do we get the vector in the direction of a with length compab?

A.We need to multiply the unit vector in the direction of a by compab.

aa

ba

a

a

a

babproja

2

We call this the vector projection of b onto a

Examples/Practice!

Key Points

• Vector algebra: addition, subtraction, scalar multiplication

• Geometric interpretation• Unit vectors• The dot product and the angle between

vectors• Projections (algebraic and geometric)