**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.)**

Answered: Image /qna-images/answer/23469964-8b1f-4a3d-9998-f4dab6b61927.jpg

Answered: 3 Compute x using the vectors w = -4…

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

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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.)**

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