Chapter 9: Q. 52 (page 631)

In Problems 50–52, determine whether v and w are parallel, orthogonal, or neither.
$v = 3 i - 2 j; w = 4 i + 6 j$

Short Answer

Expert verified

The given vectors are orthogonal but not parallel.

Step by step solution

Step 1. Given Information

The given vectors are $v = 3 i - 2 j a n d w = 4 i + 6 j .$

Step 2. Determining whether v and w are parallel or not

To determine whether the two vectors are parallel or not, we have to find the angle between them because two vectors v and w are said to be parallel if the angle between them is $0 o r π .$

Let's find the angle

$\cos θ = \frac{v \cdot w}{||v|| ||w||} \cos θ = \frac{(3 i - 2 j) \cdot (4 i + 6 j)}{(\sqrt{3^{2} + {(- 2)}^{2}}) (\sqrt{{(4)}^{2} + {(6)}^{2}})} \cos θ = \frac{0}{26} \cos θ = 0 θ = \cos^{- 1} (0) θ = \frac{π}{2}$

Thus, the given vectors are not parallel because the angle is neither $0 n o r π .$

Step 3. Determining whether v and w are orthogonal or not

Two vectors v and w are said to be orthogonal if and only if $v \cdot w = 0 .$

Let's find the dot product

$v \cdot w = (3 i - 2 j) \cdot (4 i + 6 j) v \cdot w = 12 - 12 v \cdot w = 0$

Thus, the vectors are orthogonal.

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In Problems 50–52, determine whether v and w are parallel, orthogonal, or neither.
$v = 3 i - 2 j; w = 4 i + 6 j$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Determining whether v and w are parallel or not

Step 3. Determining whether v and w are orthogonal or not

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In Problems 50–52, determine whether v and w are parallel, orthogonal, or neither.v=3i-2j;w=4i+6j

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Determining whether v and w are parallel or not

Step 3. Determining whether v and w are orthogonal or not

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Discrete Mathematics

Decision Maths

Probability and Statistics

Applied Mathematics

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

In Problems 50–52, determine whether v and w are parallel, orthogonal, or neither.
$v = 3 i - 2 j; w = 4 i + 6 j$