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Mr. Simonds MTH 253

D o t a n d C r o s s P r o d u c t s S e c t i o n s 9 . 3 a n d 9 . 4 | 1

Example 1

Find the dot product between the vectors 0,3=u and 2,2=v .

Example 2

Use the Law of Cosines to establish that the dot product between the vectors 21 , uuu = and

21 , vvv = is equal to 2211 vuvu + . (In linear algebra, this latter quantity is called the inner

product of u and v .)

Figure 1 : 0,3=u and 2,2=v . u

v

45

B ( )21 , vv

O ( )0,0

A ( )21 , uu

22

21 vvv +=

22

21 uuu +=

( ) ( )2 21 1 2 2v u v u= + AB

Figure 2: 21 , uuu = and 21 , vvv = ; A , B , and O are points.

Law of Cosines: ( )2 2 2

2 cosAB u v u v = +

Definition

( )cosu v u v =

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Mr. Simonds MTH 253

D o t a n d C r o s s P r o d u c t s S e c t i o n s 9 . 3 a n d 9 . 4 | 3

Example 5

Constance has got herself quite the boyfriend. The emergency brake on her 3000 lb car busted afew weeks ago; this was a major problem because Constance lives on a street with a 10% grade.Luckily for Constance, her hunky sweetie Bruno volunteered to act as a hitching post and is alwaysavailable to have the car strapped to his back when Constance is at home; or wherever.

Lets determine just how much force Bruno has to counteract when Constances car is strapped tohis back. Lets do so in a way that the concepts of projection vector and component are cleverlyintroduced.

So Constance, wheresBruno this evening?

Oh, he had otherimportant things toattend to tonight.

proj v u

u

Let v be any vector that points down the hill.

Common tail pointfor all three vector

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Mr. Simonds MTH 253

4 | D o t a n d C r o s s P r o d u c t s S e c t i o n s 9 . 3 - 9 . 4

So does it make any difference if the vector being projected onto points down or up the hill?

Turns out that the answer is no and yes. Curses!

Example 6

Find, and illustrate, proj v u where 5, 7u = and 4, 1v = . First find compv u .

Example 7

Find, and illustrate, proj w u where 5, 7u = and 4,1w = . First find comp v u .

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Mr. Simonds MTH 253

6 | D o t a n d C r o s s P r o d u c t s S e c t i o n s 9 . 3 - 9 . 4

Example 9: Projections put to geometric use

Find the shortest distance between the parallel lines x y 24 = and x y 23 = .

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Mr. Simonds MTH 253

D o t a n d C r o s s P r o d u c t s S e c t i o n s 9 . 3 a n d 9 . 4 | 7

Cross Product

Example 9

Find the cross products u v

and uv . State the geometric relationship between the cross

products and each vector in the cross product. Show, too please, that ( )sinu v u v =

where is the smaller angle formed when u and v are drawn tail-to-tail.

3,2,1=u and 5,6,7=v

Example 10

Find u v

without a lick of calculator work. Then verify your result with a lick or two of calculator

work. Oh yeah 4,2=u and 7,1=v .

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Mr. Simonds MTH 253

8 | D o t a n d C r o s s P r o d u c t s S e c t i o n s 9 . 3 - 9 . 4

Example 11

Find the equation for the plane that passes through the points ( ): 2, 5, 6A , ( ): 2,5, 4B and( ): 1, 7, 5C .