Chapter 9: Q. 19 (page 612)

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

Short Answer

Expert verified

$v_{1} = 2.5 i - 2.5 j v_{2} = - 0.5 i - 0.5 j$

Step by step solution

Step 1. Concept Used

The decomposition of $v$ into $v_{1}$ and localid="1646825977563" $v_{2}$ , where $v_{1}$ is parallel to $w$ and $v_{2}$ is orthogonal to $w$ , is

localid="1646826554773" $v_{1} = \frac{v \cdot w}{{|w|}^{2}} w v_{2} = v - v_{1}$

Step 2. Find the dot product and w

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

$v \cdot w = (2 i - 3 j) \cdot (i - j) v \cdot w = 2 \cdot 1 + (- 3) \cdot (- 1) v \cdot w = 2 + 3 v \cdot w = 5$

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

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

Step 3. Find v1 and v2

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

role="math" $v_{1} = \frac{v \cdot w}{{|w|}^{2}} w$

On substituting the calculated values we get

$v_{1} = \frac{5}{{(\sqrt{2})}^{2}} (i - j) v_{1} = \frac{5}{2} (i - j) v_{1} = 2.5 i - 2.5 j$

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

$v_{2} = v - v_{1} v_{2} = 2 i - 3 j - (2.5 i - 2.5 j) v_{2} = 2 i - 3 j - 2.5 i + 2.5 j v_{2} = (2 - 2.5) i + (- 3 + 2.5) j v_{2} = - 0.5 i - 0.5 j$

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

Short Answer

Step by step solution

Step 1. Concept Used

Step 2. Find the dot product and w

Step 3. Find v1 and v2

One App. One Place for Learning.

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In Problems 19–24, decompose v into two vectors v1and v2, where v1 is parallel to w and v2 is orthogonal to w.v=2i-3jw=i-j

Short Answer

Step by step solution

Step 1. Concept Used

Step 2. Find the dot product and w

Step 3. Find v1 and v2

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Probability and Statistics

Discrete Mathematics

Logic and Functions

Applied Mathematics

Study anywhere. Anytime. Across all devices.

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