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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. 24 (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.
$v = i - 3 j, w = 4 i - j$

Short Answer

Expert verified

The values are $v_{1} = \frac{28}{17} i - \frac{7}{17} j$ and $v_{2} = - \frac{11}{17} i - \frac{44}{17} j .$

Step by step solution

01

Step 1. Given Information

The two vectors are $v = i - 3 j a n d w = 4 i - j .$

We have to decomposevinto two vectors $v_{1} a n d v_{2},$ where $v_{1}$ is parallel to wand $v_{2}$ is orthogonal tow.

02

Step 2. Finding v1

To find $v_{1},$ we will use the formula $v_{1} = \frac{v 路 w}{{||w||}^{2}} w$

Now,

$v 路 w = (i - 3 j) 路 (4 i - j) v 路 w = 4 + 3 v 路 w = 7$

And

$||w|| = \sqrt{{(4)}^{2} + {(- 1)}^{2}} ||w|| = \sqrt{16 + 1} ||w|| = \sqrt{17} {||w||}^{2} = 17$

Let's put all the values in the formula

$v_{1} = \frac{7}{17} (4 i - j) v_{1} = \frac{28}{17} i - \frac{7}{17} j$

03

Step 3. Finding v2

To find $v_{2}$ use the formula $v_{2} = v - v_{1}$

So,

$v_{2} = (i - 3 j) - (\frac{28}{17} i - \frac{7}{17} j) v_{2} = (1 - \frac{28}{17}) i + (- 3 + \frac{7}{17}) j v_{2} = - \frac{11}{17} i - \frac{44}{17} j$

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