6.4 VECTORS AND DOT PRODUCTS

Copyright © Cengage Learning. All rights reserved.

2© 2015 by Michael R. Green

How do you write v in terms of sine and cosine?

cos , sin or cos sin v v i v j

We will be using this fact again in this section.

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• Find the dot product of two vectors and use the properties of the dot product.

• Find the angle between two vectors and determine whether two vectors are orthogonal.

• Write a vector as the sum of two vector components.

• Use vectors to find the work done by a force.

What You Should Learn Sec 6-4

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The Dot Product of Two Vectors

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The Dot Product of Two Vectors

In this section, you will study a third vector operation, the dot product. This product yields a scalar, rather than a vector.

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The Dot Product of Two Vectors

Proofs of these properties can be found on page 490 of your text.

v v v

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Example 1 – Finding Dot Products

Find each dot product.

a. 4, 5 2, 3 b. 2, –1 1, 2 c. 0, 3 4, –2

Solution:

a. 4, 5 2, 3 = 4(2) + 5(3)

= 8 + 15

= 23

b. 2, –1 1, 2 = 2(1) + (–1)(2)

= 2 – 2

= 0

Go to LIST

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Example 1 – Solution

c. 0, 3 4, –2 = 0(4) + 3(–2)

= 0 – 6

= – 6

cont’d

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Complete section 1 NTG, page 119

On the TI-36X Pro:1. Go to vector, arrow

over to edit, enter the 1st vector. Repeat for the 2nd vector.

2. Go to vector, arrow over to math, choose DotProduct, go back to vector, choose each vector, separated by a comma.

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You Do Exercise 7, page 465

Both the TI-84 and the TI-36X Pro require more button pushing than just using the definition.

11Part a ends up being a scalar times a vector, b is the dot product of 2 vectors.

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You Do Exercise 17, page 465.

Find :u v

Now find the scalar product:

_24, 12 ; vector

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You Do Exercise 25

2

_ _8, 15 8, 15

u u u

u

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The Angle Between Two Vectors

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The Angle Between Two Vectors

The angle between two nonzerovectors is the angle , 0 ,between their respective standard position vectors, as shown in

Figure 6.33. This angle can be foundusing the dot product.

Figure 6.33

A proof may be found on page 490 of your text.

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Example 4 – Finding the Angle Between Two Vectors

Find the angle between u = 4, 3 and v = 3, 5.

Solution:

The two vectors and are shown in Figure 6.34.

Figure 6.34

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Example 4 – Solution

This implies that the angle between the two vectors is

cont’d

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You Do Exercise 35, page 465

_

22 _ 2 2

2, 1 6, 4

2 1 6 4

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The Angle Between Two Vectors

Rewriting the expression for the angle between two vectors in the form

u v = || u || || v || cos

produces an alternative way to calculate the dot product.

From this form, you can see that because || u || and || v || are always positive, u v and cos will always have the same sign.

Alternative form of dot product

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The Angle Between Two Vectors

Figure 6.35 shows the five possible orientations of two vectors.

Figure 6.35

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The Angle Between Two Vectors

This is a direct result of the previous formula with case 3.

The terms orthogonal and perpendicular mean essentially the same thing—meeting at right angles.

Note that the zero vector is orthogonal to every vector u, because 0 u = 0.

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You Do Exercise 53, page 466.

The vectors are parallel (and travelling in opposite directions).This is case 1 from page 460.

If we were only determining if they are orthogonal, the dot product would be enough.