Let's calculate each part step by step.

Given:

• $u = \langle 4, 8, -8 \rangle$
• $v = \langle 9, -10, 11 \rangle$ (from $9i - 10j + 11k$)
• $w = \langle 5, 0, -2 \rangle$ (from $5i - 2k$)

a) Calculate $3u - 4v$

1. Calculate $3u$: $3u = 3 \langle 4, 8, -8 \rangle = \langle 12, 24, -24 \rangle$

2. Calculate $4v$: $4v = 4 \langle 9, -10, 11 \rangle = \langle 36, -40, 44 \rangle$

3. Now subtract: $3u - 4v = \langle 12, 24, -24 \rangle - \langle 36, -40, 44 \rangle = \langle 12 - 36, 24 + 40, -24 - 44 \rangle = \langle -24, 64, -68 \rangle$

b) Calculate $|u + w|$

1. Calculate $u + w$: $u + w = \langle 4, 8, -8 \rangle + \langle 5, 0, -2 \rangle = \langle 4 + 5, 8 + 0, -8 - 2 \rangle = \langle 9, 8, -10 \rangle$

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

c) Calculate $v \cdot w$

1. Calculate the dot product: $v \cdot w = \langle 9, -10, 11 \rangle \cdot \langle 5, 0, -2 \rangle = 9 \times 5 + (-10) \times 0 + 11 \times (-2) = 45 + 0 - 22 = 23$

d) Find a unit vector parallel to $u$

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

2. The unit vector $\hat{u}$ parallel to $u$ is given by: $\hat{u} = \frac{u}{|u|} = \frac{\langle 4, 8, -8 \rangle}{12} = \langle \frac{1}{3}, \frac{2}{3}, -\frac{2}{3} \rangle$

Summary of Results:

• a) $3u - 4v = \langle -24, 64, -68 \rangle$
• b) $|u + w| = 7\sqrt{5}$
• c) $v \cdot w = 23$
• d) A unit vector parallel to $u = \langle \frac{1}{3}, \frac{2}{3}, -\frac{2}{3} \rangle$

Would you like any details or further explanations on these calculations? Here are some related questions you might find interesting:

1. How would you calculate the cross product of $u$ and $v$?
2. What is the angle between vectors $v$ and $w$?
3. How do you find the projection of $v$ onto $w$?
4. What happens to the unit vector if $u$ is scaled by a factor of 2?
5. How do you verify if two vectors are orthogonal?

Tip: Remember that the dot product of two vectors is zero if they are orthogonal (perpendicular to each other).