Let's solve the given vector operations step by step.

Given vectors:

$\mathbf{a} = (-3, -4, 7)$ $\mathbf{b} = (8, -9, 8)$ $\mathbf{c} = (6, -4, -8)$ $\mathbf{d} = (3, 8, -6)$

1. Compute $\mathbf{a} - \mathbf{c}$:

$\mathbf{a} - \mathbf{c} = (-3, -4, 7) - (6, -4, -8)$

$= (-3 - 6, -4 - (-4), 7 - (-8))$

$= (-9, 0, 15)$

2. Compute $\mathbf{a} + \mathbf{b} - \mathbf{d}$:

$\mathbf{a} + \mathbf{b} - \mathbf{d} = (-3, -4, 7) + (8, -9, 8) - (3, 8, -6)$

Step by step:

$(-3 + 8, -4 + (-9), 7 + 8)$

$= (5, -13, 15)$

Now subtract $\mathbf{d}$:

$(5 - 3, -13 - 8, 15 - (-6))$

$= (2, -21, 21)$

3. Compute $\mathbf{a} + \mathbf{b} - \mathbf{c} - \mathbf{d}$:

$\mathbf{a} + \mathbf{b} - \mathbf{c} - \mathbf{d} = (-3, -4, 7) + (8, -9, 8) - (6, -4, -8) - (3, 8, -6)$

First, compute $\mathbf{a} + \mathbf{b}$:

$(-3 + 8, -4 + (-9), 7 + 8)$

$= (5, -13, 15)$

Now subtract $\mathbf{c}$:

$(5 - 6, -13 - (-4), 15 - (-8))$

$= (-1, -9, 23)$

Now subtract $\mathbf{d}$:

$(-1 - 3, -9 - 8, 23 - (-6))$

$= (-4, -17, 29)$

Final Answers:

$\mathbf{a} - \mathbf{c} = (-9, 0, 15)$

$\mathbf{a} + \mathbf{b} - \mathbf{d} = (2, -21, 21)$

$\mathbf{a} + \mathbf{b} - \mathbf{c} - \mathbf{d} = (-4, -17, 29)$

Would you like a breakdown of any step?

Here are five related questions to deepen understanding:

1. How do you compute the sum of two vectors geometrically?
2. What is the significance of vector subtraction in physics?
3. How can vector operations be applied in 3D graphics?
4. What is the difference between a scalar and a vector quantity?
5. How do you find the magnitude of a vector?

Tip:

Always double-check your vector operations by verifying each coordinate separately to avoid sign errors.