Chapter 9: Q. 53 (page 631)

In Problems 53 and 54, decompose v into two vectors, one parallel to w and the other orthogonal to w.
$v = 2 i + j; w = - 4 i + 3 j$

Short Answer

Expert verified

The two vectors are $v_{1} = \frac{4}{5} i - \frac{3}{5} j and v_{2} = \frac{6}{5} i + \frac{8}{5} j .$

Step by step solution

Step 1. Given Information

The given vectors are $v = 2 i + j a n d w = - 4 i + 3 j .$

Step 2. Finding the vector parallel to w

Let $v_{1}$ be the vector parallel to w.To find $v_{1}$ , we will use formula $v_{1} = \frac{v \cdot w}{{||w||}^{2}} w .$

First, find the dot product

role="math" localid="1647003393909" $v \cdot w = (2 i + j) \cdot (- 4 i + 3 j) v \cdot w = - 8 + 3 v \cdot w = - 5$

Now, let's find ${||w||}^{2}$

$||w|| = \sqrt{{(- 4)}^{2} + 3^{2}} ||w|| = \sqrt{25} ||w|| = 5 {||w||}^{2} = 25$

Put all the values in the formula

$v_{1} = \frac{- 5}{25} (- 4 i + 3 j) v_{1} = \frac{20}{25} i - \frac{15}{25} j v_{1} = \frac{4}{5} i - \frac{3}{5} j$

Step 3. Finding the vector orthogonal to w

Let $v_{2}$ be the vector orthogonal to w.To find $v_{2}$ we will use the formula $v_{2} = v - v_{1} .$

So,

$v_{2} = (2 i + j) - (\frac{4}{5} i - \frac{3}{5} j) v_{2} = \frac{6}{5} i + \frac{8}{5} j$

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In Problems 53 and 54, decompose v into two vectors, one parallel to w and the other orthogonal to w.
$v = 2 i + j; w = - 4 i + 3 j$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the vector parallel to w

Step 3. Finding the vector orthogonal to w

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In Problems 53 and 54, decompose v into two vectors, one parallel to w and the other orthogonal to w.v=2i+j;w=-4i+3j

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the vector parallel to w

Step 3. Finding the vector orthogonal to w

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Logic and Functions

Applied Mathematics

Probability and Statistics

Theoretical and Mathematical Physics

Pure Maths

Study anywhere. Anytime. Across all devices.

In Problems 53 and 54, decompose v into two vectors, one parallel to w and the other orthogonal to w.
$v = 2 i + j; w = - 4 i + 3 j$