3 Compute x using the vectors w = -4 and x = 2 - 5 2 x = (Simplify your answer. Type an integer or simplified fraction for each matrix element.)
3 Compute x using the vectors w = -4 and x = 2 - 5 2 x = (Simplify your answer. Type an integer or simplified fraction for each matrix element.)
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![**Vectors and Matrix Operations**
To compute the vector projection of vector **x** onto vector **w**, we utilize the formula:
\[ \frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w} \]
Given the vectors:
\[ \mathbf{w} = \begin{bmatrix} 3 \\ -4 \\ -5 \end{bmatrix} \]
\[ \mathbf{x} = \begin{bmatrix} 6 \\ -2 \\ 2 \end{bmatrix} \]
The form of the computation we will use is:
\[ \frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{x} \cdot \mathbf{x}} \mathbf{x} \]
We start by computing the dot products:
1. **Calculate \( \mathbf{x} \cdot \mathbf{w} \):**
\[
(6)(3) + (-2)(-4) + (2)(-5) = 18 + 8 - 10 = 16
\]
2. **Calculate \( \mathbf{x} \cdot \mathbf{x} \):**
\[
(6)(6) + (-2)(-2) + (2)(2) = 36 + 4 + 4 = 44
\]
Now, substituting back into the equation:
\[
\frac{16}{44} \mathbf{x}
\]
Simplify the fraction \( \frac{16}{44} \):
\[
\frac{4}{11} \mathbf{x}
\]
Thus calculating the components:
\[
\frac{4}{11} \begin{bmatrix} 6 \\ -2 \\ 2 \end{bmatrix} = \begin{bmatrix} \frac{24}{11} \\ -\frac{8}{11} \\ \frac{8}{11} \end{bmatrix}
\]
The result of the operation is:
\[
\begin{pmatrix} \frac{24}{11} \\ -\frac{8}{11} \\ \frac{8}{11} \end{pmatrix}
\]
**(Simplify your answer. Type an integer or simplified fraction for each matrix element.)**](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f9df24d-bd71-4c3f-9ec7-f4cd302c8c7d%2F23469964-8b1f-4a3d-9998-f4dab6b61927%2Fq2xvx6w_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Vectors and Matrix Operations**
To compute the vector projection of vector **x** onto vector **w**, we utilize the formula:
\[ \frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} \mathbf{w} \]
Given the vectors:
\[ \mathbf{w} = \begin{bmatrix} 3 \\ -4 \\ -5 \end{bmatrix} \]
\[ \mathbf{x} = \begin{bmatrix} 6 \\ -2 \\ 2 \end{bmatrix} \]
The form of the computation we will use is:
\[ \frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{x} \cdot \mathbf{x}} \mathbf{x} \]
We start by computing the dot products:
1. **Calculate \( \mathbf{x} \cdot \mathbf{w} \):**
\[
(6)(3) + (-2)(-4) + (2)(-5) = 18 + 8 - 10 = 16
\]
2. **Calculate \( \mathbf{x} \cdot \mathbf{x} \):**
\[
(6)(6) + (-2)(-2) + (2)(2) = 36 + 4 + 4 = 44
\]
Now, substituting back into the equation:
\[
\frac{16}{44} \mathbf{x}
\]
Simplify the fraction \( \frac{16}{44} \):
\[
\frac{4}{11} \mathbf{x}
\]
Thus calculating the components:
\[
\frac{4}{11} \begin{bmatrix} 6 \\ -2 \\ 2 \end{bmatrix} = \begin{bmatrix} \frac{24}{11} \\ -\frac{8}{11} \\ \frac{8}{11} \end{bmatrix}
\]
The result of the operation is:
\[
\begin{pmatrix} \frac{24}{11} \\ -\frac{8}{11} \\ \frac{8}{11} \end{pmatrix}
\]
**(Simplify your answer. Type an integer or simplified fraction for each matrix element.)**
Expert Solution
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 3 steps with 3 images
Knowledge Booster
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
