Let's calculate each of the parts step by step.

Given:

• $\mathbf{u} = \begin{pmatrix} 4 \\ 8 \\ -8 \end{pmatrix}$
• $\mathbf{v} = \begin{pmatrix} 0 \\ -10 \\ 11 \end{pmatrix}$
• $\mathbf{w} = \begin{pmatrix} 5 \\ 0 \\ -2 \end{pmatrix}$

a) $3\mathbf{u} - 4\mathbf{v}$

Calculating $3\mathbf{u}$: $3\mathbf{u} = 3 \begin{pmatrix} 4 \\ 8 \\ -8 \end{pmatrix} = \begin{pmatrix} 12 \\ 24 \\ -24 \end{pmatrix}$

Calculating $4\mathbf{v}$: $4\mathbf{v} = 4 \begin{pmatrix} 0 \\ -10 \\ 11 \end{pmatrix} = \begin{pmatrix} 0 \\ -40 \\ 44 \end{pmatrix}$

Now, subtract $4\mathbf{v}$ from $3\mathbf{u}$: $3\mathbf{u} - 4\mathbf{v} = \begin{pmatrix} 12 \\ 24 \\ -24 \end{pmatrix} - \begin{pmatrix} 0 \\ -40 \\ 44 \end{pmatrix} = \begin{pmatrix} 12 \\ 64 \\ -68 \end{pmatrix}$

b) $|\mathbf{u} + \mathbf{w}|$

Calculating $\mathbf{u} + \mathbf{w}$: $\mathbf{u} + \mathbf{w} = \begin{pmatrix} 4 \\ 8 \\ -8 \end{pmatrix} + \begin{pmatrix} 5 \\ 0 \\ -2 \end{pmatrix} = \begin{pmatrix} 9 \\ 8 \\ -10 \end{pmatrix}$

Now, calculating the magnitude: $|\mathbf{u} + \mathbf{w}| = \sqrt{9^2 + 8^2 + (-10)^2} = \sqrt{81 + 64 + 100} = \sqrt{245} = 7\sqrt{5}$

c) $\mathbf{v} \cdot \mathbf{w}$

Calculating the dot product: $\mathbf{v} \cdot \mathbf{w} = \begin{pmatrix} 0 \\ -10 \\ 11 \end{pmatrix} \cdot \begin{pmatrix} 5 \\ 0 \\ -2 \end{pmatrix} = (0)(5) + (-10)(0) + (11)(-2) = 0 + 0 - 22 = -22$

d) A unit vector parallel to $\mathbf{u}$

First, find the magnitude of $\mathbf{u}$: $|\mathbf{u}| = \sqrt{4^2 + 8^2 + (-8)^2} = \sqrt{16 + 64 + 64} = \sqrt{144} = 12$

Now, the unit vector $\mathbf{u}_{unit}$: $\mathbf{u}_{unit} = \frac{1}{|\mathbf{u}|} \mathbf{u} = \frac{1}{12} \begin{pmatrix} 4 \\ 8 \\ -8 \end{pmatrix} = \begin{pmatrix} \frac{1}{3} \\ \frac{2}{3} \\ -\frac{2}{3} \end{pmatrix}$

Summary of Results:

a) $3\mathbf{u} - 4\mathbf{v} = \begin{pmatrix} 12 \\ 64 \\ -68 \end{pmatrix}$
b) $|\mathbf{u} + \mathbf{w}| = 7\sqrt{5}$
c) $\mathbf{v} \cdot \mathbf{w} = -22$
d) A unit vector parallel to $\mathbf{u} = \begin{pmatrix} \frac{1}{3} \\ \frac{2}{3} \\ -\frac{2}{3} \end{pmatrix}$

Do you want details on any part or have any questions? Here are some related questions to consider:

1. How would the results change if we used different vectors for $\mathbf{u}$, $\mathbf{v}$, and $\mathbf{w}$?
2. Can you explain how to calculate the cross product of two vectors?
3. What is the geometric interpretation of the dot product?
4. How can you verify if two vectors are orthogonal?
5. What are some applications of unit vectors in physics?

Tip: When working with vectors, always check your calculations for accuracy, especially with sign and magnitude.