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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. 17 (page 632)

In Problems 14鈥�18, $P_{1} = (3 \sqrt{2}, 7 \sqrt{2})$ and $P_{2} = (8 \sqrt{2}, 2 \sqrt{2})$ .
Find the direction angle of $v$ .

Short Answer

Expert verified

The directional angles are $伪 = 45 掳尾 = 315 掳$

Step by step solution

01

Step 1. Given information

The points are

$P_{1} = (3 \sqrt{2}, 7 \sqrt{2}) P_{2} = (8 \sqrt{2}, 2 \sqrt{2})$

02

Step 2. Find the direction angle of v.

Position vector $v$ is equal to $\overset{鈫�}{P_{1} P_{2}}$

$\overset{鈫�}{P_{1} P_{2}} = \overset{鈫�}{O P_{2}} - \overset{鈫�}{O P_{1}} = (8 \sqrt{2} \hat{i} + 2 \sqrt{2} \hat{j}) - (3 \sqrt{2} \hat{i} + 7 \sqrt{2} \hat{j}) = 5 \sqrt{2} \hat{i} - 5 \sqrt{2} \hat{j}$

The direction angles $伪, 尾$ of $v = a \hat{i} + b \hat{j}$ are given by

localid="1647012506438" $\cos 伪 = \frac{a}{\sqrt{a^{2} + b^{2}}} = \frac{5 \sqrt{2}}{\sqrt{{(5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{2}}} = \frac{1}{\sqrt{2}} 伪 = 45 掳 \cos 尾 = \frac{b}{\sqrt{a^{2} + b^{2}}} = \frac{- 5 \sqrt{2}}{\sqrt{{(5 \sqrt{2})}^{2} + {(- 5 \sqrt{2})}^{2}}} = - \frac{1}{\sqrt{2}} 尾 = 315 掳$

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