Chapter 9: Q. 49 (page 630)

In Problems 46–49, find the dot product $v \cdot w$ and the angle between v and w.
$v = 4 i - j + 2 k, w = i - 2 j - 3 k$

Short Answer

Expert verified

The dot product $v \cdot w$ is $0$ and the angle between v and w is $90 ° .$

Step by step solution

Step 1. Given Information

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

Step 2. Finding the dot product

To find the dot product, use the formula of the dot product which is $v \cdot w = a_{1} a_{2} + b_{1} b_{2} .$

So,

$v \cdot w = (4) (1) + (- 1) (- 2) + (2) (- 3) v \cdot w = 4 + 2 - 6 v \cdot w = 0$

Step 3. Finding the angle between v and w

To find the angle between vectors, we will use the formula $\cos θ = \frac{u \cdot v}{||u|| ||v||}$

We have the dot product but to find the length of a vector we will use $||u|| = \sqrt{a^{2} + b^{2}} .$

So,

$\cos θ = \frac{(4 i - j + 2 k) \cdot (i - 2 j - 3 k)}{(\sqrt{4^{2} + {(- 1)}^{2} + 2^{2}}) (\sqrt{1^{2} + {(- 2)}^{2} + {(- 3)}^{2}})} \cos θ = \frac{0}{(\sqrt{4^{2} + {(- 1)}^{2} + 2^{2}}) (\sqrt{1^{2} + {(- 2)}^{2} + {(- 3)}^{2}})} θ = \cos^{- 1} (0) θ = 90 °$

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In Problems 46–49, find the dot product $v \cdot w$ and the angle between v and w.
$v = 4 i - j + 2 k, w = i - 2 j - 3 k$

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the dot product

Step 3. Finding the angle between v and w

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In Problems 46–49, find the dot product v·w and the angle between v and w. v=4i-j+2k,w=i-2j-3k

Short Answer

Step by step solution

Step 1. Given Information

Step 2. Finding the dot product

Step 3. Finding the angle between v and w

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Logic and Functions

Pure Maths

Calculus

Mechanics Maths

Discrete Mathematics

Study anywhere. Anytime. Across all devices.

In Problems 46–49, find the dot product $v \cdot w$ and the angle between v and w.
$v = 4 i - j + 2 k, w = i - 2 j - 3 k$