Find the following using u = <4, 8, −8>, v = 9i − 10j + 11k, and w = 5i − 2k.
a) 3u − 4v
b) |u + w|
c) v · w
d) A unit vector parallel to u

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).

Find the following using u = <4, 8, −8>, v = 9i − 10j + 11k, and w = 5i − 2k: a) 3u − 4v b) |u + w| c) v · w d) A unit vector parallel to u.

Explore how to calculate 3u - 4v, the magnitude |u + w|, the dot product v · w, and find a unit vector parallel to u using the vectors u = <4, 8, −8>, v = 9i − 10j + 11k, and w = 5i − 2k. This guide is designed for students in Grades 9-12.

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