### Vector Operations Exercise**Problem 2:**Compute $\mathbf{w} \cdot \mathbf{w}$, $\mathbf{x} \cdot \mathbf{w}$, and $\frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}}$ using the vectors $\mathbf{w}$ and $\mathbf{x}$.Given vectors:\[\mathbf{w} = \begin{bmatrix}4 \\-1 \\-5\end{bmatrix}, \quad \text{and} \quad \mathbf{x} = \begin{bmatrix}7 \\-3 \\3\end{bmatrix}\]### Tasks:1. Calculate $\mathbf{w} \cdot \mathbf{w}$ \[ \mathbf{w} \cdot \mathbf{w} = \underline{\hspace{2cm}} \] *(Simplify your answer. Type an integer or a simplified fraction.)*2. Calculate $\mathbf{x} \cdot \mathbf{w}$ \[ \mathbf{x} \cdot \mathbf{w} = \underline{\hspace{2cm}} \] *(Simplify your answer. Type an integer or a simplified fraction.)*3. Calculate $\frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}}$ \[ \frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} = \underline{\hspace{2cm}} \] *(Simplify your answer. Type an integer or a simplified fraction.)*

Answered: Image /qna-images/answer/3c48a832-5bb8-4fe2-a1f4-6f85c0b13e7c.jpg

2. 4 7 X•w Compute w • w, x•w, and using the vectors w=-1 and x = w•w - 5 3 w•w = (Simplify your answer. Type an integer or a simplified fraction.) x•w = (Simplify your answer. Type an integer or a simplified fraction.) w•w (Simplify your answer. Type an integer or a simplified fraction.) 3.

2. 4 7 X•w Compute w • w, x•w, and using the vectors w=-1 and x = w•w - 5 3 w•w = (Simplify your answer. Type an integer or a simplified fraction.) x•w = (Simplify your answer. Type an integer or a simplified fraction.) w•w (Simplify your answer. Type an integer or a simplified fraction.) 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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Question

### Vector Operations Exercise

**Problem 2:**

Compute $\mathbf{w} \cdot \mathbf{w}$, $\mathbf{x} \cdot \mathbf{w}$, and $\frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}}$ using the vectors $\mathbf{w}$ and $\mathbf{x}$.

Given vectors:
\[
\mathbf{w} =
\begin{bmatrix}
4 \\
-1 \\
-5
\end{bmatrix},
\quad \text{and} \quad
\mathbf{x} =
\begin{bmatrix}
7 \\
-3 \\
3
\end{bmatrix}
\]

### Tasks:

1. Calculate $\mathbf{w} \cdot \mathbf{w}$
\[
\mathbf{w} \cdot \mathbf{w} = \underline{\hspace{2cm}}
\]
*(Simplify your answer. Type an integer or a simplified fraction.)*

2. Calculate $\mathbf{x} \cdot \mathbf{w}$
\[
\mathbf{x} \cdot \mathbf{w} = \underline{\hspace{2cm}}
\]
*(Simplify your answer. Type an integer or a simplified fraction.)*

3. Calculate $\frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}}$
\[
\frac{\mathbf{x} \cdot \mathbf{w}}{\mathbf{w} \cdot \mathbf{w}} = \underline{\hspace{2cm}}
\]
*(Simplify your answer. Type an integer or a simplified fraction.)*

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