Chapter 9: Q. 54 (page 631)

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

Short Answer

Expert verified

The two vectors are $v_{1} = \frac{27}{10} i + \frac{9}{10} j a n d v_{2} = - \frac{7}{10} i + \frac{21}{10} j .$

Step by step solution

Step 1. Given Information

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

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

$v \cdot w = (2 i + 3 j) \cdot (3 i + j) v \cdot w = 6 + 3 v \cdot w = 9$

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

$||w|| = \sqrt{{(3)}^{2} + 1^{2}} ||w|| = \sqrt{10} {||w||}^{2} = 10$

Put all the values in the formula

$v_{1} = \frac{9}{10} (3 i + j) v_{1} = \frac{27}{10} i + \frac{9}{10} j$

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,

$v_{2} = (2 i + 3 j) - (\frac{27}{10} i + \frac{9}{10} j) v_{2} = - \frac{7}{10} i + \frac{21}{10} j$

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

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+3j;w=3i+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

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Probability and Statistics

Geometry

Pure Maths

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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.
$v = 2 i + 3 j; w = 3 i + j$