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 $\frac{5}{9} [\begin{array}{l} 2 \\ 1 \\ 2 \end{array}]$ .

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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

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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

Probability and Statistics

Theoretical and Mathematical Physics

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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.