Algebra and Trigonometry (6th Edition)
6th Edition
ISBN:9780134463216
Author:Robert F. Blitzer
Publisher:Robert F. Blitzer
ChapterP: Prerequisites: Fundamental Concepts Of Algebra
Section: Chapter Questions
Problem 1MCCP: In Exercises 1-25, simplify the given expression or perform the indicated operation (and simplify,...
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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 \([c_1, c_2, c_3]\):](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcc6ab625-9bc8-400b-b1f4-23740d203705%2Fef05963a-46d4-46d4-9745-fd6f84562f97%2F8yhxuw_processed.png&w=3840&q=75)
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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