Q7E 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: Q7E (page 71)

Let Lbe the line in $R^{3}$ that consists of all scalar multiples of localid="1664190410191" $[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$ . Find the reflection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ about the line L.

Short Answer

Expert verified

The reflection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ about the line L is $\frac{1}{9} [\begin{array}{l} 11 \\ 1 \\ 11 \end{array}]$ .

Step by step solution

Reflection 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 linear transformation $T (\overset{鈫�}{x}) = {\overset{鈫�}{x}}^{鈭�} 鈭� {\overset{鈫�}{x}}^{鈯�}$ is called the reflection of about L, often denoted by $r e f_{L} (\overset{鈫�}{x})$ :

$\begin{array}{l} r e f_{L} (\overset{鈫�}{x}) = {\overset{鈫�}{x}}^{鈭�} 鈭� {\overset{鈫�}{x}}^{鈯�} \\ r e f_{L} (\overset{鈫�}{x}) = 2 (\overset{鈫�}{x} . \overset{鈫�}{u}) \overset{鈫�}{u} 鈭� \overset{鈫�}{x} \end{array}$

Finding reflection 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 $\overset{鈫�}{u}$

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

Now put the values in the formula $r e f_{L} (\overset{鈫�}{x}) = 2 (\overset{鈫�}{x} . \overset{鈫�}{u}) \overset{鈫�}{u} 鈭� \overset{鈫�}{x}$ , we will get.

$\begin{array}{l} r e f_{L} (\overset{鈫�}{x}) = 2 ([\begin{array}{l} 1 \\ 1 \\ 1 \end{array}] . \frac{1}{3} [\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]) \frac{1}{3} [\begin{array}{l} 2 \\ 1 \\ 2 \end{array}] 鈭� [\begin{array}{l} 1 \\ 1 \\ 1 \end{array}] \\ r e f_{L} (\overset{鈫�}{x}) = \frac{10}{9} [\begin{array}{l} 2 \\ 1 \\ 2 \end{array}] 鈭� [\begin{array}{l} 1 \\ 1 \\ 1 \end{array}] = \frac{1}{9} [\begin{array}{l} 11 \\ 1 \\ 11 \end{array}] \end{array}$

Hence, the reflection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ about the line L is $\frac{1}{9} [\begin{array}{l} 11 \\ 1 \\ 11 \end{array}]$ .

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91影视

Let Lbe the line in $R^{3}$ that consists of all scalar multiples of localid="1664190410191" $[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$ . Find the reflection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ about the line L.

Short Answer

Step by step solution

Reflection of vector

Finding reflection of vector

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91影视

Let Lbe the line in R3that consists of all scalar multiples of localid="1664190410191" [212]. Find the reflection of the vector[111]about the line L.

Short Answer

Step by step solution

Reflection of vector

Finding reflection of vector

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Statistics

Mechanics Maths

Pure Maths

Geometry

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Let Lbe the line in $R^{3}$ that consists of all scalar multiples of localid="1664190410191" $[\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$ . Find the reflection of the vector $[\begin{array}{l} 1 \\ 1 \\ 1 \end{array}]$ about the line L.