Consider the following vectors: --8-8-8 H W1 = -3 W2 = -1 4 Enter the vector n in the form [C₁, C₂, C3]: = The set B = {w₁, W2} is an orthogonal basis of a subspace W = Span (w₁, W₂) of R³. Find a vector n which is orthogonal to W, and such that vn is in W.
Consider the following vectors: --8-8-8 H W1 = -3 W2 = -1 4 Enter the vector n in the form [C₁, C₂, C3]: = The set B = {w₁, W2} is an orthogonal basis of a subspace W = Span (w₁, W₂) of R³. Find a vector n which is orthogonal to W, and such that vn is in W.
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
![Consider the following vectors:
\[
\mathbf{w}_1 = \begin{bmatrix} -1 \\ -3 \\ 1 \end{bmatrix}, \quad \mathbf{w}_2 = \begin{bmatrix} 7 \\ -1 \\ 4 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} -4 \\ -4 \\ -1 \end{bmatrix}
\]
The set \(\mathcal{B} = \{\mathbf{w}_1, \mathbf{w}_2\}\) is an orthogonal basis of a subspace \(W = \text{Span}(\mathbf{w}_1, \mathbf{w}_2)\) of \(\mathbb{R}^3\). Find a vector \(\mathbf{n}\) which is orthogonal to \(W\), and such that \(\mathbf{v} - \mathbf{n}\) is in \(W\).
Enter the vector \(\mathbf{n}\) in the form \([c_1, c_2, c_3]\):
\[ \begin{bmatrix} \hspace{1cm} \hspace{1cm} \hspace{1cm} \end{bmatrix} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb493ecdd-cbdc-400d-a05a-de2010eb2d52%2Fad6af361-3985-483a-8064-01eb6987d081%2F2slbrl_processed.png&w=3840&q=75)
Transcribed Image Text:Consider the following vectors:
\[
\mathbf{w}_1 = \begin{bmatrix} -1 \\ -3 \\ 1 \end{bmatrix}, \quad \mathbf{w}_2 = \begin{bmatrix} 7 \\ -1 \\ 4 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} -4 \\ -4 \\ -1 \end{bmatrix}
\]
The set \(\mathcal{B} = \{\mathbf{w}_1, \mathbf{w}_2\}\) is an orthogonal basis of a subspace \(W = \text{Span}(\mathbf{w}_1, \mathbf{w}_2)\) of \(\mathbb{R}^3\). Find a vector \(\mathbf{n}\) which is orthogonal to \(W\), and such that \(\mathbf{v} - \mathbf{n}\) is in \(W\).
Enter the vector \(\mathbf{n}\) in the form \([c_1, c_2, c_3]\):
\[ \begin{bmatrix} \hspace{1cm} \hspace{1cm} \hspace{1cm} \end{bmatrix} \]
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