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Precalculus Enhanced with Graphing Utilities

Sullivan

$Math Studyset 91影视 Explanations$ Math

6th Edition 路 1200 pages

ISBN: 9780321795465

Chapter 9: Q. 49 (page 630)

In Problems 46鈥�49, find the dot product $v 路 w$ and the angle between v and w.
$v = 4 i - j + 2 k, w = i - 2 j - 3 k$

Short Answer

Expert verified

The dot product $v 路 w$ is $0$ and the angle between v and w is $90 掳 .$

Step by step solution

01

Step 1. Given Information

The given vectors are $v = 4 i - j + 2 k a n d w = i - 2 j - 3 k .$

02

Step 2. Finding the dot product

To find the dot product, use the formula of the dot product which is $v 路 w = a_{1} a_{2} + b_{1} b_{2} .$

So,

$v 路 w = (4) (1) + (- 1) (- 2) + (2) (- 3) v 路 w = 4 + 2 - 6 v 路 w = 0$

03

Step 3. Finding the angle between v and w

To find the angle between vectors, we will use the formula $\cos 胃 = \frac{u 路 v}{||u|| ||v||}$

We have the dot product but to find the length of a vector we will use $||u|| = \sqrt{a^{2} + b^{2}} .$

So,

$\cos 胃 = \frac{(4 i - j + 2 k) 路 (i - 2 j - 3 k)}{(\sqrt{4^{2} + {(- 1)}^{2} + 2^{2}}) (\sqrt{1^{2} + {(- 2)}^{2} + {(- 3)}^{2}})} \cos 胃 = \frac{0}{(\sqrt{4^{2} + {(- 1)}^{2} + 2^{2}}) (\sqrt{1^{2} + {(- 2)}^{2} + {(- 3)}^{2}})} 胃 = \cos^{- 1} (0) 胃 = 90 掳$

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Most popular questions from this chapter

Use the figure at the right. Determine whether the given statement is true or false.

$E + D = G + H$

Identify and graph each polar equation. Verify your graph using a graphing utility.

$r = 1 - \cos 胃$

Use the vectors in the figure at the right to graph each of the following vectors.

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$(0, 2)$

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