Let y = 3 H -3 -2 and u = projuy = 2 1 (a) Find the orthogonal projection of y onto u. H (b) Compute the distance d from y to the line through u and the origin. d=
Let y = 3 H -3 -2 and u = projuy = 2 1 (a) Find the orthogonal projection of y onto u. H (b) Compute the distance d from y to the line through u and the origin. d=
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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![### Linear Algebra Problem: Orthogonal Projection and Distance Calculation
Given the vectors:
\[ \mathbf{y} = \begin{bmatrix} 3 \\ -2 \\ -3 \end{bmatrix} \quad \text{and} \quad \mathbf{u} = \begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix} \]
(a) **Find the orthogonal projection of \(\mathbf{y}\) onto \(\mathbf{u}\).**
\[ \text{proj}_{\mathbf{u}}\mathbf{y} = \begin{bmatrix}
\boxed{} \\ \boxed{} \\ \boxed{}
\end{bmatrix} \]
(b) **Compute the distance \(d\) from \(\mathbf{y}\) to the line through \(\mathbf{u}\) and the origin.**
\[ d = \boxed{} \]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffb2133c9-e1e5-4d56-9c72-044227328930%2Fe1847d97-fe23-4515-a619-16a78b88d33e%2F3owzwlp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Linear Algebra Problem: Orthogonal Projection and Distance Calculation
Given the vectors:
\[ \mathbf{y} = \begin{bmatrix} 3 \\ -2 \\ -3 \end{bmatrix} \quad \text{and} \quad \mathbf{u} = \begin{bmatrix} 2 \\ 1 \\ -1 \end{bmatrix} \]
(a) **Find the orthogonal projection of \(\mathbf{y}\) onto \(\mathbf{u}\).**
\[ \text{proj}_{\mathbf{u}}\mathbf{y} = \begin{bmatrix}
\boxed{} \\ \boxed{} \\ \boxed{}
\end{bmatrix} \]
(b) **Compute the distance \(d\) from \(\mathbf{y}\) to the line through \(\mathbf{u}\) and the origin.**
\[ d = \boxed{} \]
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