Find x so that the dot product of the vectors u and v below is 23.
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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Question
![To solve the problem of finding \( x \) so that the dot product of the vectors \( \mathbf{u} \) and \( \mathbf{v} \) is 23, we are given the following vectors:
\[
\mathbf{u} = \begin{bmatrix} 3 \\ 2 \\ x \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 5 \\ -2 \\ 2 \end{bmatrix}
\]
The dot product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is calculated as:
\[
\mathbf{u} \cdot \mathbf{v} = (3)(5) + (2)(-2) + (x)(2)
\]
Simplifying the expression, we get:
\[
15 - 4 + 2x = 23
\]
Further simplification:
\[
11 + 2x = 23
\]
Subtracting 11 from both sides:
\[
2x = 12
\]
Dividing by 2:
\[
x = 6
\]
However, it seems the solution mentioned in the image states \( x = 0 \), which is incorrect based on the dot product being equal to 23. By recalculating correctly, \( x = 6 \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fec2b7ff9-b952-4215-9e0f-f264e2036fb8%2Fbe9f89f2-8ae2-480c-975d-07a28fb56851%2Frzmr02h_processed.png&w=3840&q=75)
Transcribed Image Text:To solve the problem of finding \( x \) so that the dot product of the vectors \( \mathbf{u} \) and \( \mathbf{v} \) is 23, we are given the following vectors:
\[
\mathbf{u} = \begin{bmatrix} 3 \\ 2 \\ x \end{bmatrix}, \quad \mathbf{v} = \begin{bmatrix} 5 \\ -2 \\ 2 \end{bmatrix}
\]
The dot product of two vectors \( \mathbf{u} \) and \( \mathbf{v} \) is calculated as:
\[
\mathbf{u} \cdot \mathbf{v} = (3)(5) + (2)(-2) + (x)(2)
\]
Simplifying the expression, we get:
\[
15 - 4 + 2x = 23
\]
Further simplification:
\[
11 + 2x = 23
\]
Subtracting 11 from both sides:
\[
2x = 12
\]
Dividing by 2:
\[
x = 6
\]
However, it seems the solution mentioned in the image states \( x = 0 \), which is incorrect based on the dot product being equal to 23. By recalculating correctly, \( x = 6 \).
Expert Solution
Step 1
dot product will be 3*5+2*(-2)+2x
so this must be 23
Step by step
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