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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. 20 (page 612)

In Problems 19鈥�24, decompose v into two vectors $v_{1}$ and $v_{2}$ , where $v_{1}$ is parallel to w and $v_{2}$ is orthogonal to w.
$v = - 3 i + 2 j w = 2 i + j$

Short Answer

Expert verified

$v_{1} = - 1.6 i - 0.8 j v_{2} = - 1.4 i + 2.8 j$

Step by step solution

01

Step 1. Concept Used

The decomposition of $v$ into $v_{1}$ and $v_{2}$ , where $v_{1}$ is parallel to $w$ and $v_{2}$ is orthogonal to $w$ , is

role="math" localid="1646826707527" $v_{1} = \frac{v 路 w}{{|w|}^{2}} w v_{2} = v - v_{1}$

02

Step 2. Find the dot product and w

For the vectors $v = - 3 i + 2 j$ and $w = 2 i + j$ the dot product is given as

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

The magnitude of vector $w = 2 i + j$ is given as

$|w| = \sqrt{2^{2} + 1^{2}} |w| = \sqrt{4 + 1} |w| = \sqrt{5}$

03

Step 3. Find v1 and v2

The vector $v_{1}$ which is parallel to $w$ is given by the formula

$v_{1} = \frac{v 路 w}{{|w|}^{2}} w$

On substituting the calculated values we get

$v_{1} = \frac{- 4}{{(\sqrt{5})}^{2}} (2 i + j) v_{1} = \frac{- 4}{5} (2 i + j) v_{1} = - 0.8 (2 i + j) v_{1} = - 1.6 i - 0.8 j$

and the vector $v_{2}$ orthogonal to $w$ is found as

$v_{2} = v - v_{1} v_{2} = - 3 i + 2 j - (- 1.6 i - 0.8 j) v_{2} = - 3 i + 2 j + 1.6 i + 0.8 j v_{2} = (- 3 + 1.6) i + (2 + 0.8) j v_{2} = 1.4 i + 2.8 j$

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Most popular questions from this chapter

If $\overset{鈬赌}{v} = 2 \overset{鈬赌}{i} - \overset{鈬赌}{j}$ and $\overset{鈬赌}{w} = x \overset{鈬赌}{i} + 3 \overset{鈬赌}{j}$ , find all numbers x for which $|\overset{鈬赌}{v} + \overset{鈬赌}{w}| = 5$ .

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