I see the set of vector problems written in the image. Let's break down each problem step by step and provide a detailed solution. I'll start with problem 42 and continue through each one.

Problem 42:

Given two points:

• $A(3, -4, 1)$
• $B(4, 6, -3)$

Vector $\mathbf{b} = (1, 3, -4)$.

You are asked to:

• (a) Find $\mathbf{a} = \overrightarrow{AB}$
• (b) Find $\mathbf{C} = 3\mathbf{a} - 2\mathbf{b}$
• (c) Find $\mathbf{d} = \mathbf{b} + 5\mathbf{a}$, and calculate the magnitude $|\mathbf{d}|$.

Problem 42 (a):

First, calculate vector $\mathbf{a} = \overrightarrow{AB}$:

$\mathbf{a} = B - A = (4 - 3, 6 - (-4), -3 - 1) = (1, 10, -4)$

So, $\mathbf{a} = (1, 10, -4)$.

Problem 42 (b):

Now, calculate $\mathbf{C} = 3\mathbf{a} - 2\mathbf{b}$.

$3\mathbf{a} = 3(1, 10, -4) = (3, 30, -12)$ $2\mathbf{b} = 2(1, 3, -4) = (2, 6, -8)$ $\mathbf{C} = (3, 30, -12) - (2, 6, -8) = (3 - 2, 30 - 6, -12 + 8) = (1, 24, -4)$

So, $\mathbf{C} = (1, 24, -4)$.

Problem 42 (c):

Finally, calculate $\mathbf{d} = \mathbf{b} + 5\mathbf{a}$.

$5\mathbf{a} = 5(1, 10, -4) = (5, 50, -20)$ $\mathbf{d} = (1, 3, -4) + (5, 50, -20) = (1 + 5, 3 + 50, -4 - 20) = (6, 53, -24)$

Now, find the magnitude $|\mathbf{d}|$:

$|\mathbf{d}| = \sqrt{6^2 + 53^2 + (-24)^2} = \sqrt{36 + 2809 + 576} = \sqrt{3421}$

So, $|\mathbf{d}| \approx 58.48$.

Problem 43:

You are asked to find the coordinates of vector $\mathbf{a}$, given $|\mathbf{a}| = 3$ and the angle ratios $\alpha = \beta = 2\gamma$.

This involves using vector magnitude and directional cosines. Let me know if you'd like a detailed solution for this next problem or any other!

Would you like to proceed with the next problem or need more clarifications on any part of this solution?

Here are 5 related questions for you:

1. How can we generalize vector subtraction to higher dimensions?
2. What is the geometric interpretation of multiplying a vector by a scalar (like $3\mathbf{a}$)?
3. Can two different vectors result in the same magnitude but different directions?
4. How does the dot product help in understanding vector angles?
5. How would the magnitude change if we altered one component of $\mathbf{d}$?

Tip: Always check units and dimensions when performing vector calculations, as they help ensure that results make sense geometrically.