127 (68) -100 -4 23 B = v= 27 is an orthogonal basis of R³. Find [v]⁄·
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
![Given:
\[
\mathcal{B} = \left\{ \begin{bmatrix} 127 \\ -100 \\ -4 \end{bmatrix}, \begin{bmatrix} 4 \\ 5 \\ 2 \end{bmatrix}, \begin{bmatrix} -4 \\ -6 \\ 23 \end{bmatrix} \right\}
\]
is an orthogonal basis of \(\mathbb{R}^3\). Find \([\mathbf{v}]_{\mathcal{B}}\).
\[
\mathbf{v} = \begin{bmatrix} 2 \\ 7 \\ 3 \end{bmatrix}
\]
Explanation:
The problem is to find the vector \( \mathbf{v} \) expressed in the orthogonal basis \( \mathcal{B} \). The basis \( \mathcal{B} \) consists of three vectors, which are orthogonal (each vector is perpendicular to the others), making calculations simpler. Expressing a vector in terms of an orthogonal basis involves calculating the dot products of \( \mathbf{v} \) with each basis vector and dividing by the magnitude squared of each basis vector. These calculations will yield the coordinates of \( \mathbf{v} \) in terms of the orthogonal basis \( \mathcal{B} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F807728d5-81cc-4dac-b6a1-9785980c085b%2F4db3b459-bfb8-4664-8687-24681159c0a2%2Fam1vq9u_processed.png&w=3840&q=75)
Transcribed Image Text:Given:
\[
\mathcal{B} = \left\{ \begin{bmatrix} 127 \\ -100 \\ -4 \end{bmatrix}, \begin{bmatrix} 4 \\ 5 \\ 2 \end{bmatrix}, \begin{bmatrix} -4 \\ -6 \\ 23 \end{bmatrix} \right\}
\]
is an orthogonal basis of \(\mathbb{R}^3\). Find \([\mathbf{v}]_{\mathcal{B}}\).
\[
\mathbf{v} = \begin{bmatrix} 2 \\ 7 \\ 3 \end{bmatrix}
\]
Explanation:
The problem is to find the vector \( \mathbf{v} \) expressed in the orthogonal basis \( \mathcal{B} \). The basis \( \mathcal{B} \) consists of three vectors, which are orthogonal (each vector is perpendicular to the others), making calculations simpler. Expressing a vector in terms of an orthogonal basis involves calculating the dot products of \( \mathbf{v} \) with each basis vector and dividing by the magnitude squared of each basis vector. These calculations will yield the coordinates of \( \mathbf{v} \) in terms of the orthogonal basis \( \mathcal{B} \).
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