Let's break down the problem and solve it step by step.

(a) Finding the Magnitude and Direction of $\mathbf{A} + \mathbf{B}$

Given:

• Vector $\mathbf{A}$ has a magnitude of 63 units and points due west.
• Vector $\mathbf{B}$ has a magnitude of 63 units and points due south.

Step 1: Represent the vectors in component form

1. Vector $\mathbf{A}$:
   Since $\mathbf{A}$ points due west, it has no vertical component (i.e., its $y$-component is 0), and its $x$-component is $-63$ (negative because it points west).
   $\mathbf{A} = \langle -63, 0 \rangle$

2. Vector $\mathbf{B}$:
   Since $\mathbf{B}$ points due south, it has no horizontal component (i.e., its $x$-component is 0), and its $y$-component is $-63$ (negative because it points south).
   $\mathbf{B} = \langle 0, -63 \rangle$

Step 2: Add the vectors

The resultant vector $\mathbf{R} = \mathbf{A} + \mathbf{B}$ is calculated by adding the corresponding components of $\mathbf{A}$ and $\mathbf{B}$: [ \mathbf{R} = \mathbf{A} + \mathbf{B} = \langle -63, 0 \