Algebra and Trigonometry (6th Edition)
Algebra and Trigonometry (6th Edition)
6th Edition
ISBN: 9780134463216
Author: Robert F. Blitzer
Publisher: PEARSON
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Question
Consider the following vectors: \[ \mathbf{w}_1 = \begin{bmatrix} -1 \\ -2 \\ 1 \end{bmatrix}, \quad \mathbf{w}_2 = \begin{bmatrix} -8 \\ 5 \\ 2 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} -3 \\ 0 \\ -4 \end{bmatrix} \] 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 \([c_1, c_2, c_3]\):
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Transcribed Image Text:Consider the following vectors: \[ \mathbf{w}_1 = \begin{bmatrix} -1 \\ -2 \\ 1 \end{bmatrix}, \quad \mathbf{w}_2 = \begin{bmatrix} -8 \\ 5 \\ 2 \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} -3 \\ 0 \\ -4 \end{bmatrix} \] 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 \([c_1, c_2, c_3]\):
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