Chapter 9: Q. 20 (page 632)

In Problems 19–22, $v_{1} = <4, 6>, v_{2} = <- 3, - 6>, v_{3} = <- 8, 4>, v_{4} = <10, 15>$
Which two vectors are parallel?

Short Answer

Expert verified

$v_{1} a n d v_{4}$ are parallel.

Step by step solution

Step 1. Given information

The given vectors are
$v_{1} = <4, 6>, v_{2} = <- 3, - 6>, v_{3} = <- 8, 4>, v_{4} = <10, 15>$

Step 2. Find the vectors which are parallel.

For finding vectors which are parallel, the angle between vectors should be $0$ or $180 °$ .

We will find the angles between vectors.

For finding the angles between any two vectors $\vec{a}$ and $\vec{b}$ ,

localid="1646912519577" $\cos θ = \frac{\vec{a} \cdot \vec{b}}{||\vec{a}|| ||\vec{b}||}$

Step 3. Find the angle between v1 and v2.

$\cos θ = \frac{v_{1} \cdot v_{2}}{||v_{1}|| ||v_{2}||} = \frac{- 12 + (- 36)}{(\sqrt{4^{2} + 6^{2}}) (\sqrt{{(- 3)}^{2} + {(- 6)}^{2}})} = \frac{- 48}{\sqrt{52} \times \sqrt{45}}$

These are not parallel.

Step 4. Find the angle between v2 and v3.

$\cos θ = \frac{v_{2} \cdot v_{3}}{||v_{2}|| ||v_{3}||} = \frac{24 + (- 24)}{(\sqrt{{(- 8)}^{2} + 4^{2}}) (\sqrt{{(- 3)}^{2} + {(- 6)}^{2}})} = 0 θ = 90 °$

These are not parallel.

Step 5. Find the angle between v3 and v4.

$\cos θ = \frac{v_{3} \cdot v_{4}}{||v_{3}|| ||v_{4}||} = \frac{- 80 + 60}{(\sqrt{{(- 8)}^{2} + 4^{2}}) (\sqrt{10^{2} + 15^{2}})} = \frac{- 20}{\sqrt{80} \times \sqrt{325}}$

These are not parallel.

Step 6. Find the angle between v1 and v4.

$\cos θ = \frac{v_{1} \cdot v_{4}}{||v_{1}|| ||v_{4}||} = \frac{40 + 90}{(\sqrt{4^{2} + 6^{2}}) (\sqrt{10^{2} + 15^{2}})} = \frac{130}{\sqrt{52} \times \sqrt{325}} = 1 θ = 0 °$

These are parallel.

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In Problems 19–22, $v_{1} = <4, 6>, v_{2} = <- 3, - 6>, v_{3} = <- 8, 4>, v_{4} = <10, 15>$
Which two vectors are parallel?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the vectors which are parallel.

Step 3. Find the angle between v1 and v2.

Step 4. Find the angle between v2 and v3.

Step 5. Find the angle between v3 and v4.

Step 6. Find the angle between v1 and v4.

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In Problems 19–22, v1=4,6,v2=-3,-6,v3=-8,4,v4=10,15Which two vectors are parallel?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the vectors which are parallel.

Step 3. Find the angle between v1 and v2.

Step 4. Find the angle between v2 and v3.

Step 5. Find the angle between v3 and v4.

Step 6. Find the angle between v1 and v4.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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Study anywhere. Anytime. Across all devices.

In Problems 19–22, $v_{1} = <4, 6>, v_{2} = <- 3, - 6>, v_{3} = <- 8, 4>, v_{4} = <10, 15>$
Which two vectors are parallel?