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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. 55 (page 631)

Find the direction angles of the vector $v = 3 i - 4 j + 2 k .$

Short Answer

Expert verified

The direction angles of the vector are $伪鈮� 56.1 掳, 尾鈮� 138 掳, a n d 纬鈮� 68.2 掳 .$

Step by step solution

01

Step 1. Given Information

The given vector is $v = 3 i - 4 j + 2 k .$

02

Step 2. Finding the direction angles

To find the direction angles $伪, 尾, a n d 纬$ use the formula

$\cos 伪 = \frac{a}{||v||}, \cos 尾 = \frac{b}{||v||}, \cos 纬 = \frac{c}{||v||} w h e r e ||v|| = \sqrt{a^{2} + b^{2} + c^{2}}$

Let's first find $||v||$

So, $||v|| = \sqrt{3^{2} + {(- 4)}^{2} + 2^{2}} ||v|| = \sqrt{29}$

03

Step 3. Solve

Proceed with the calculation further by using the formula.

Thus,

$\cos 伪 = \frac{3}{\sqrt{29}} Rationalize the denominator 肠辞蝉伪 = \frac{3 \sqrt{29}}{29} 伪 = \cos^{- 1} (\frac{3 \sqrt{29}}{29}) 伪鈮� 56.1 掳$

And

$\cos 尾 = \frac{- 4}{\sqrt{29}} Rationalize the denominator 肠辞蝉尾 = - \frac{4 \sqrt{29}}{29} 尾 = \cos^{- 1} (- \frac{4 \sqrt{29}}{29}) 尾鈮� 138 掳$

And

$\cos 纬 = \frac{2}{\sqrt{29}} Rationalize the denominator 肠辞蝉纬 = \frac{2 \sqrt{29}}{29} 纬 = \cos^{- 1} (\frac{2 \sqrt{29}}{29}) 纬鈮� 68.2 掳$

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