Chapter 10: Q 21. (page 812)

In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.
$u = <2, 0, - 5>, v = <- 3, 7, - 1>$

Short Answer

Expert verified

The dot product is $- 1$ and the angle isrole="math" localid="1649664356753" $\cos^{- 1} (\frac{- 1}{\sqrt{1711}})$ .

Step by step solution

Step 1. Given information.

The given pairs of vectors are:

$u = <2, 0, - 5> and v = <- 3, 7, - 1>$

Step 2. Find the dot product.

The dot product is: $u \cdot v = u_{1} v_{1} + u_{2} v_{2} + u_{3} v_{3}$

$u = <2, 0, - 5> and v = <- 3, 7, - 1> = 2 (- 3) + 0 (7) + (- 5) (- 1) = - 6 + 0 + 5 = - 1$

Therefore, the dot product of the given two vectors $u = <2, 0, - 5> and v = <- 3, 7, - 1>$ is $- 1$ .

Step 3. Find the angle between the two vectors.

The formula for the angle between the two vectors is:

$\cos θ = \frac{u \cdot v}{||u|| ||v||}$

$u = <2, 0, - 5> and v = <- 3, 7, - 1> u \cdot v = - 1 ||u|| = \sqrt{2^{2} + 0^{2} + {(- 5)}^{2}} \sqrt{4 + 0 + 25} = \sqrt{29} ||v|| = \sqrt{{(- 3)}^{2} + 7^{2} + {(- 1)}^{2}} \sqrt{9 + 49 + 1} = \sqrt{59}$

Then,

role="math" localid="1649664305708" $\cos θ = \frac{u \cdot v}{||u|| ||v||} \cos θ = \frac{- 1}{\sqrt{29} \sqrt{59}} θ = \cos^{- 1} (\frac{- 1}{\sqrt{1711}})$

Therefore, the angle isrole="math" localid="1649664318896" $\cos^{- 1} (\frac{- 1}{\sqrt{1711}})$ .

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In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.
$u = <2, 0, - 5>, v = <- 3, 7, - 1>$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the dot product.

Step 3. Find the angle between the two vectors.

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In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.u=2,0,-5,v=-3,7,-1

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the dot product.

Step 3. Find the angle between the two vectors.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Applied Mathematics

Theoretical and Mathematical Physics

Mechanics Maths

Statistics

Geometry

Probability and Statistics

Study anywhere. Anytime. Across all devices.

In Exercises 20-23, find the dot product of the given pairs of vectors and the angle between the two vectors.
$u = <2, 0, - 5>, v = <- 3, 7, - 1>$