Find a vector that is orthogonal to each of the three vectors below: V1 = " V2 = 0 and V3 = 1 3 3
Find a vector that is orthogonal to each of the three vectors below: V1 = " V2 = 0 and V3 = 1 3 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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Can you please do this the matrix way because I am desperate please this has to be done using the matrix way and can you do it step by step
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## Orthogonal Vector Problem
### Problem Statement
1. **Find a vector that is orthogonal to each of the three vectors below:**
\[ v_1 = \begin{bmatrix} 1 \\ 1 \\ 4 \\ -1 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \quad \text{and} \quad v_3 = \begin{bmatrix} -1 \\ 1 \\ 3 \\ 3 \end{bmatrix}. \]
### Explanation
To solve this problem, you need to find a vector \( \mathbf{u} \) such that it is perpendicular (orthogonal) to all given vectors \( \mathbf{v_1} \), \( \mathbf{v_2} \), and \( \mathbf{v_3} \). This means that the dot product of \( \mathbf{u} \) with each of these vectors should equal zero.
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This content will help students understand the concept of orthogonality in the context of vectors and how to apply mathematical principles to find a common orthogonal vector in multiple dimensions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F94fb8ffe-bd98-40e6-b1c3-546ba39a0620%2Ffc831d41-040d-44ce-a53e-0d043c205ec6%2Fcaacn2u_processed.jpeg&w=3840&q=75)
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## Orthogonal Vector Problem
### Problem Statement
1. **Find a vector that is orthogonal to each of the three vectors below:**
\[ v_1 = \begin{bmatrix} 1 \\ 1 \\ 4 \\ -1 \end{bmatrix}, \quad v_2 = \begin{bmatrix} 1 \\ 0 \\ 1 \\ 0 \end{bmatrix}, \quad \text{and} \quad v_3 = \begin{bmatrix} -1 \\ 1 \\ 3 \\ 3 \end{bmatrix}. \]
### Explanation
To solve this problem, you need to find a vector \( \mathbf{u} \) such that it is perpendicular (orthogonal) to all given vectors \( \mathbf{v_1} \), \( \mathbf{v_2} \), and \( \mathbf{v_3} \). This means that the dot product of \( \mathbf{u} \) with each of these vectors should equal zero.
---
This content will help students understand the concept of orthogonality in the context of vectors and how to apply mathematical principles to find a common orthogonal vector in multiple dimensions.
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