Refer to the vectors u, to ug. 5. U1 = u2 = uz = u4 = Us = U6 = uz = ug = 4 -1 Determine if the given vectors are orthogonal. (a) uz and uz uz and uz are orthogonal. u2 and uz are not orthogonal. (b) uj and u2 uj and uz are orthogonal. uj and uz are not orthogonal. (c) ug and us ug and ug are orthogonal. ug and us are not orthogonal. (d) uz and u6 uz and ug are orthogonal. uz and ug are not orthogonal.
Refer to the vectors u, to ug. 5. U1 = u2 = uz = u4 = Us = U6 = uz = ug = 4 -1 Determine if the given vectors are orthogonal. (a) uz and uz uz and uz are orthogonal. u2 and uz are not orthogonal. (b) uj and u2 uj and uz are orthogonal. uj and uz are not orthogonal. (c) ug and us ug and ug are orthogonal. ug and us are not orthogonal. (d) uz and u6 uz and ug are orthogonal. uz and ug are not orthogonal.
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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![**Vectors and Orthogonality**
Below are vectors \( \mathbf{u}_1 \) to \( \mathbf{u}_8 \).
\[
\mathbf{u}_1 = \begin{bmatrix} -5 \\ 1 \\ 4 \end{bmatrix}, \quad \mathbf{u}_2 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{u}_3 = \begin{bmatrix} 4 \\ 0 \\ -1 \end{bmatrix}, \quad \mathbf{u}_4 = \begin{bmatrix} 1 \\ -5 \\ 4 \end{bmatrix}, \quad \mathbf{u}_5 = \begin{bmatrix} 4 \\ 1 \\ 1 \end{bmatrix}
\]
\[
\mathbf{u}_6 = \begin{bmatrix} 0 \\ 5 \\ 1 \end{bmatrix}, \quad \mathbf{u}_7 = \begin{bmatrix} -7 \\ 5 \\ -4 \end{bmatrix}, \quad \mathbf{u}_8 = \begin{bmatrix} -1 \\ -1 \\ 5 \end{bmatrix}
\]
**Determine if the given vectors are orthogonal:**
(a) \( \mathbf{u}_2 \) and \( \mathbf{u}_3 \)
- ☐ \( \mathbf{u}_2 \) and \( \mathbf{u}_3 \) are orthogonal.
- ☐ \( \mathbf{u}_2 \) and \( \mathbf{u}_3 \) are not orthogonal.
(b) \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \)
- ☐ \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) are orthogonal.
- ☐ \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) are not orthogonal.
(c) \( \mathbf{u}_8 \) and \( \mathbf{u}_5 \)
- ☐ \( \mathbf{u}_8 \) and \( \mathbf{u}_5 \) are orthogonal.
- ☐ \( \mathbf{u}_8 \) and \( \mathbf{u}_5 \) are not orthogonal.
(d)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad35cd49-88f1-4759-9e4c-232c0c792d3e%2Fa0f86ff0-d55f-499e-a8b1-d9538e4357a0%2F5b8e3qo_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vectors and Orthogonality**
Below are vectors \( \mathbf{u}_1 \) to \( \mathbf{u}_8 \).
\[
\mathbf{u}_1 = \begin{bmatrix} -5 \\ 1 \\ 4 \end{bmatrix}, \quad \mathbf{u}_2 = \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}, \quad \mathbf{u}_3 = \begin{bmatrix} 4 \\ 0 \\ -1 \end{bmatrix}, \quad \mathbf{u}_4 = \begin{bmatrix} 1 \\ -5 \\ 4 \end{bmatrix}, \quad \mathbf{u}_5 = \begin{bmatrix} 4 \\ 1 \\ 1 \end{bmatrix}
\]
\[
\mathbf{u}_6 = \begin{bmatrix} 0 \\ 5 \\ 1 \end{bmatrix}, \quad \mathbf{u}_7 = \begin{bmatrix} -7 \\ 5 \\ -4 \end{bmatrix}, \quad \mathbf{u}_8 = \begin{bmatrix} -1 \\ -1 \\ 5 \end{bmatrix}
\]
**Determine if the given vectors are orthogonal:**
(a) \( \mathbf{u}_2 \) and \( \mathbf{u}_3 \)
- ☐ \( \mathbf{u}_2 \) and \( \mathbf{u}_3 \) are orthogonal.
- ☐ \( \mathbf{u}_2 \) and \( \mathbf{u}_3 \) are not orthogonal.
(b) \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \)
- ☐ \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) are orthogonal.
- ☐ \( \mathbf{u}_1 \) and \( \mathbf{u}_2 \) are not orthogonal.
(c) \( \mathbf{u}_8 \) and \( \mathbf{u}_5 \)
- ☐ \( \mathbf{u}_8 \) and \( \mathbf{u}_5 \) are orthogonal.
- ☐ \( \mathbf{u}_8 \) and \( \mathbf{u}_5 \) are not orthogonal.
(d)
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