12 2 Determine if the pair of vectors v₁ = 3 V₂ = =-3 are orthogonal. -5 3
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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![**Problem Statement:**
Determine if the pair of vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \) are orthogonal.
Given:
\[ \mathbf{v_1} = \begin{bmatrix} 12 \\ 3 \\ -5 \end{bmatrix}, \ \mathbf{v_2} = \begin{bmatrix} 2 \\ -3 \\ 3 \end{bmatrix} \]
**Solution:**
Two vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \) are orthogonal if their dot product is zero. The dot product of two vectors \( \mathbf{v_1} = \begin{bmatrix} v_{11} \\ v_{12} \\ v_{13} \end{bmatrix} \) and \( \mathbf{v_2} = \begin{bmatrix} v_{21} \\ v_{22} \\ v_{23} \end{bmatrix} \) is given by:
\[ \mathbf{v_1} \cdot \mathbf{v_2} = v_{11}v_{21} + v_{12}v_{22} + v_{13}v_{23} \]
Substituting the given vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \):
\[ \mathbf{v_1} \cdot \mathbf{v_2} = (12)(2) + (3)(-3) + (-5)(3) \]
\[ = 24 - 9 - 15 \]
\[ = 0 \]
Since the dot product \( \mathbf{v_1} \cdot \mathbf{v_2} \) is zero, the vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \) are orthogonal.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9baea4c0-9d9e-491a-8271-bc2a2bbd8b45%2F3e520a56-6ff2-4133-926c-ca77f47847ec%2F5lnjh7vh_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Determine if the pair of vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \) are orthogonal.
Given:
\[ \mathbf{v_1} = \begin{bmatrix} 12 \\ 3 \\ -5 \end{bmatrix}, \ \mathbf{v_2} = \begin{bmatrix} 2 \\ -3 \\ 3 \end{bmatrix} \]
**Solution:**
Two vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \) are orthogonal if their dot product is zero. The dot product of two vectors \( \mathbf{v_1} = \begin{bmatrix} v_{11} \\ v_{12} \\ v_{13} \end{bmatrix} \) and \( \mathbf{v_2} = \begin{bmatrix} v_{21} \\ v_{22} \\ v_{23} \end{bmatrix} \) is given by:
\[ \mathbf{v_1} \cdot \mathbf{v_2} = v_{11}v_{21} + v_{12}v_{22} + v_{13}v_{23} \]
Substituting the given vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \):
\[ \mathbf{v_1} \cdot \mathbf{v_2} = (12)(2) + (3)(-3) + (-5)(3) \]
\[ = 24 - 9 - 15 \]
\[ = 0 \]
Since the dot product \( \mathbf{v_1} \cdot \mathbf{v_2} \) is zero, the vectors \( \mathbf{v_1} \) and \( \mathbf{v_2} \) are orthogonal.
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