Chapter 9: Q. 20 (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.
$v = - 3 i + 2 j w = 2 i + j$

Short Answer

Expert verified

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

Step by step solution

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

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

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

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

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

Step 3. Find v1 and v2

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

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

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

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=-3i+2jw=2i+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

Pure Maths

Calculus

Probability and Statistics

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