Q6E Let L聽be the line in R3聽that c... [FREE SOLUTION]

91影视

Linear Algebra With Applications

Otto Bretscher

$Math Studyset 91影视 Explanations$ Math

5th Edition 路 442 pages

ISBN: 9780321796974

Chapter 2: Q6E (page 71)

Let Lbe the line in $R^{3}$ that consists of all scalar multiples of $[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$ . Find the orthogonal projection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ onto line L.

Short Answer

Expert verified

The orthogonal projection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ about the line L is $\frac{5}{9} [\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$ .

Step by step solution

orthogonal projection of vector

Consider a line Lin the coordinate plane, running through the origin, and let $\overset{鈫�}{x} = {\overset{鈫�}{x}}^{鈭�} 鈭� {\overset{鈫�}{x}}^{鈯�}$ be a vector in $R^{2}$ .

The transformation localid="1664191697372" $T (\overset{鈫�}{x}) = \overset{鈫�}{x}$ from $R^{2}$ to $R^{2}$ is called the orthogonal projection of $\overset{鈫�}{x}$ onto L, often denoted by $p r o j_{L} (\overset{鈫�}{x})$ .

$p r o j_{L} (\overset{鈫�}{x}) = (\overset{鈫�}{x} . \overset{鈫�}{u}) \overset{鈫�}{u}$

Finding projection of vector

Let $\overset{鈫�}{x} = [\begin{array}{l} 1 \\ 1 \\ 1 \end{array}], \overset{鈫�}{v} = [\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$

First of all, we will find the unit vector of $\overset{鈫�}{u}$

$\overset{鈫�}{u} = \frac{1}{鈥� \overset{鈫�}{v} 鈥�} \overset{鈫�}{v} = \frac{1}{3} [\begin{array}{l} 2 \\ 2 \\ 2 \end{array}]$

Now put the values in the formula $p r o j_{L} (\overset{鈫�}{x}) = (\overset{鈫�}{x} . \overset{鈫�}{u}) \overset{鈫�}{u}$ , we will get.

$\begin{array}{l} p r o j_{L} (\overset{鈫�}{x}) = ([\begin{array}{l} 1 \\ 1 \\ 1 \end{array}] . \frac{1}{3} [\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]) \frac{1}{3} [\begin{array}{l} 2 \\ 1 \\ 2 \end{array}] \\ p r o j_{L} (\overset{鈫�}{x}) = \frac{5}{9} [\begin{array}{l} 2 \\ 1 \\ 2 \end{array}] \end{array}$

Hence, the reflection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ about the line L is $\frac{5}{9} [\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$ .

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.

Over 30 million students worldwide already upgrade their learning with 91影视!

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Let Lbe the line in $R^{3}$ that consists of all scalar multiples of $[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$ . Find the orthogonal projection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ onto line L.

Short Answer

Step by step solution

orthogonal projection of vector

Finding projection of vector

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Geometry

Statistics

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

91影视

Let Lbe the line in R3that consists of all scalar multiples of[212]. Find the orthogonal projection of the vector[111]onto line L.

Short Answer

Step by step solution

orthogonal projection of vector

Finding projection of vector

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Geometry

Statistics

Pure Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

Let Lbe the line in $R^{3}$ that consists of all scalar multiples of $[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$ . Find the orthogonal projection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ onto line L.