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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. 48 (page 630)

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

Short Answer

Expert verified

The dot product $v 路 w$ is $1$ and the angle between v and w is $70.5 掳 .$

Step by step solution

01

Step 1. Given Information

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

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,

$v 路 w = (1) (1) + (1) (- 1) + (1) (1) v 路 w = 1 - 1 + 1 v 路 w = 1$

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,

$\cos 胃 = \frac{(i + j + k) 路 (i - j + k)}{(\sqrt{1^{2} + 1^{2} + 1^{2}}) (\sqrt{1^{2} + {(- 1)}^{2} + 1^{2}})} \cos 胃 = \frac{1}{\sqrt{3} \sqrt{3}} 胃 = \cos^{- 1} (\frac{1}{3}) 胃鈮� 70.5 掳$

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Most popular questions from this chapter

True or False The tests for symmetry in polar coordinates are necessary but not sufficient.

If the rectangular coordinates of a point are $(4, - 6),$ the point symmetric to it with respect to the origin is ______.

Match the graph with the polar equation.

$r = 2 \sin 胃$

find the unit vector in the same direction as $v$ $v = 3 i - 4 j$

find each quantity if $v = 3 i - 5 j$ and $w = - 2 i + 3 j$

$||v + w||$

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