Chapter 9: Q. 48 (page 630)

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

Short Answer

Expert verified

The dot product $v \cdot w$ is $1$ and the angle between v and w is $70.5 ° .$

Step by step solution

Step 1. Given Information

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

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,

$v \cdot w = (1) (1) + (1) (- 1) + (1) (1) v \cdot w = 1 - 1 + 1 v \cdot w = 1$

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,

$\cos θ = \frac{(i + j + k) \cdot (i - j + k)}{(\sqrt{1^{2} + 1^{2} + 1^{2}}) (\sqrt{1^{2} + {(- 1)}^{2} + 1^{2}})} \cos θ = \frac{1}{\sqrt{3} \sqrt{3}} θ = \cos^{- 1} (\frac{1}{3}) θ \approx 70.5 °$

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

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=i+j+k,w=i-j+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

One App. One Place for Learning.

Most popular questions from this chapter

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Applied Mathematics

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