Chapter 10: Q 31. (page 812)

In Exercises 28-31, find ${proj}_{u} v and {proj}_{v} u$ .
$u = <1, - 5, - 1>, v = <0, 1, 0>$

Short Answer

Expert verified

The values are ${proj}_{u} v = \frac{- 5}{27} and {proj}_{v} u = <0, - 5, 0>$ .

Step by step solution

Step 1. Given information.

The given vectors are:

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

Step 2. Find the vector projection of v onto u.

${proj}_{u} v = \frac{u \cdot v}{{||u||}^{2}} u u = <1, - 5, - 1> and v = <0, 1, 0> u \cdot v = 1 (0) + (- 5) 1 + (- 1) 0 = 0 - 5 + 0 = - 5 ||u|| = \sqrt{1^{2} + {(- 5)}^{2} + {(- 1)}^{2}} = \sqrt{1 + 25 + 1} = \sqrt{27}$

Then,

${proj}_{u} v = \frac{u \cdot v}{{||u||}^{2}} u = \frac{- 5}{{(\sqrt{27})}^{2}} <1, - 5, - 1> = \frac{- 5}{27} <1, - 5, - 1>$

Therefore, ${proj}_{u} v = \frac{- 5}{27} <1, - 5, - 1>$ .

Step 3. Find the vector projection of u onto v.

${proj}_{v} u = \frac{u \cdot v}{{||v||}^{2}} v u = <1, - 5, - 1> and v = <0, 1, 0> ||v|| = \sqrt{0^{2} + 1^{2} + 0^{2}} = \sqrt{1} = 1 {proj}_{v} u = \frac{u \cdot v}{{||v||}^{2}} v = \frac{- 5}{{(1)}^{2}} <0, 1, 0> = - 5 <0, 1, 0> = <0, - 5, 0>$

Therefore, ${proj}_{v} u = <0, - 5, 0>$ .

Unlock Step-by-Step Solutions & Ace Your Exams!

Full Textbook Solutions
Get detailed explanations and key concepts
Unlimited Al creation
Al flashcards, explanations, exams and more...
Ads-free access
To over 500 millions flashcards
Money-back guarantee
We refund you if you fail your exam.

Start your free trial

Over 30 million students worldwide already upgrade their learning with Vaia!

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

In Exercises 28-31, find ${proj}_{u} v and {proj}_{v} u$ .
$u = <1, - 5, - 1>, v = <0, 1, 0>$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the vector projection of v onto u.

Step 3. Find the vector projection of u onto v.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Applied Mathematics

Calculus

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

Company

Product

Help

In Exercises 28-31, find projuvandprojvu.u=1,-5,-1,v=0,1,0

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Find the vector projection of v onto u.

Step 3. Find the vector projection of u onto v.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Probability and Statistics

Applied Mathematics

Calculus

Decision Maths

Geometry

Study anywhere. Anytime. Across all devices.

In Exercises 28-31, find ${proj}_{u} v and {proj}_{v} u$ .
$u = <1, - 5, - 1>, v = <0, 1, 0>$