Chapter 9: Q. 21 (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 orthogonal?

Short Answer

Expert verified

$v_{2}$ and $v_{3}$ are orthogonal.

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 orthogonal.

For finding vectors which are parallel, the angle between vectors should be $90 °$

We will find the angles between vectors.

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

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

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 orthogonal.

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 orthogonal.

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 orthogonal.

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 not orthogonal.

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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 orthogonal?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the vectors which are orthogonal.

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 orthogonal?

Short Answer

Step by step solution

Step 1. Given information

Step 2. Find the vectors which are orthogonal.

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

Discrete Mathematics

Theoretical and Mathematical Physics

Geometry

Statistics

Logic and Functions

Decision Maths

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 orthogonal?