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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![### Problem Statement
Describe conditions on \( v = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \) such that \( \mathbf{u} \times \mathbf{v} = \mathbf{\theta} \) where \( \mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \) and \( \mathbf{\theta} \) is the zero vector.
### Explanation
To solve this problem, we need to understand the condition for the cross product of two vectors to be the zero vector. The cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is zero if and only if the vectors are parallel to each other.
Given:
\[ \mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \]
\[ \mathbf{v} = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \]
And we need to find conditions on \( a \), \( b \), and \( c \) such that
\[ \mathbf{u} \times \mathbf{v} = \mathbf{\theta} \]
### Calculation
The cross product of \( \mathbf{u} \) and \( \mathbf{v} \) is:
\[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 3 \\ a & b & c \end{vmatrix} \]
This determinant expands to:
\[ \mathbf{u} \times \mathbf{v} = \mathbf{i}(2c - 3b) - \mathbf{j}(1c - 3a) + \mathbf{k}(1b - 2a) \]
\[ \mathbf{u} \times \mathbf{v} = \begin{pmatrix} 2c - 3b \\ -c + 3a \\ b - 2a \end{pmatrix} \]
For \( \mathbf{u} \times \mathbf{v} \) to be the zero vector \( \mathbf{\theta} \), each component must be zero:
1. \( 2c - 3b = 0 \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F230d8563-210a-42c1-ba8c-cbe0d0551b01%2Fc56f42ee-a42c-48e8-a0fa-1d87f437c191%2Fx88auin_processed.png&w=3840&q=75)
Transcribed Image Text:### Problem Statement
Describe conditions on \( v = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \) such that \( \mathbf{u} \times \mathbf{v} = \mathbf{\theta} \) where \( \mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \) and \( \mathbf{\theta} \) is the zero vector.
### Explanation
To solve this problem, we need to understand the condition for the cross product of two vectors to be the zero vector. The cross product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is zero if and only if the vectors are parallel to each other.
Given:
\[ \mathbf{u} = \begin{pmatrix} 1 \\ 2 \\ 3 \end{pmatrix} \]
\[ \mathbf{v} = \begin{pmatrix} a \\ b \\ c \end{pmatrix} \]
And we need to find conditions on \( a \), \( b \), and \( c \) such that
\[ \mathbf{u} \times \mathbf{v} = \mathbf{\theta} \]
### Calculation
The cross product of \( \mathbf{u} \) and \( \mathbf{v} \) is:
\[ \mathbf{u} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 2 & 3 \\ a & b & c \end{vmatrix} \]
This determinant expands to:
\[ \mathbf{u} \times \mathbf{v} = \mathbf{i}(2c - 3b) - \mathbf{j}(1c - 3a) + \mathbf{k}(1b - 2a) \]
\[ \mathbf{u} \times \mathbf{v} = \begin{pmatrix} 2c - 3b \\ -c + 3a \\ b - 2a \end{pmatrix} \]
For \( \mathbf{u} \times \mathbf{v} \) to be the zero vector \( \mathbf{\theta} \), each component must be zero:
1. \( 2c - 3b = 0 \)
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