 Vector A has a magnitude of 63 units and points due west, while vector B has the same magnitude and points due south. Find the magnitude and direction of (a) A + B and (b) AB. Specify the directions relative to due west. ★

To solve the problem, let's break it down into two parts: (a) finding the magnitude and direction of \( \mathbf{A} + \mathbf{B} \), and (b) finding the dot product \( \mathbf{A} \cdot \mathbf{B} \).

### Part (a): Magnitude and Direction of \( \mathbf{A} + \mathbf{B} \)
Given:
- Magnitude of \( \mathbf{A} = 63 \) units, pointing due west.
- Magnitude of \( \mathbf{B} = 63 \) units, pointing due south.

Vectors \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular to each other since one points west and the other points south. To find the resultant vector \( \mathbf{A} + \mathbf{B} \), we can use the Pythagorean theorem.

1. **Magnitude of \( \mathbf{A} + \mathbf{B} \):**

\[
|\mathbf{A} + \mathbf{B}| = \sqrt{|\mathbf{A}|^2 + |\mathbf{B}|^2} = \sqrt{63^2 + 63^2} = \sqrt{2 \times 63^2} = 63\sqrt{2} \approx 89.1 \text{ units}
\]

2. **Direction of \( \mathbf{A} + \mathbf{B} \):**

The direction can be found using the tangent function, as the two vectors are perpendicular:

\[
\theta = \tan^{-1} \left(\frac{|\mathbf{B}|}{|\mathbf{A}|}\right) = \tan^{-1} \left(\frac{63}{63}\right) = \tan^{-1}(1) = 45^\circ
\]

The resultant vector \( \mathbf{A} + \mathbf{B} \) points southwest, which is \( 45^\circ \) south of west.

### Part (b): Dot Product \( \mathbf{A} \cdot \mathbf{B} \)
The dot product of two vectors is given by:

\[
\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}| \cos \theta
\]

Since \( \mathbf{A} \) and \( \mathbf{B} \) are perpendicular to each other (\( \theta = 90^\circ \)):

\[
\mathbf{A} \cdot \mathbf{B} = 63 \times 63 \times \cos(90^\circ) = 63 \times 63 \times 0 = 0
\]

### Summary of Results:
1. **Magnitude of \( \mathbf{A} + \mathbf{B} \)**: \( 89.1 \) units
2. **Direction of \( \mathbf{A} + \mathbf{B} \)**: \( 45^\circ \) south of west
3. **Dot product \( \mathbf{A} \cdot \mathbf{B} \)**: \( 0 \) (because the vectors are perpendicular)

Would you like to explore more about vector operations or have any specific questions?

Here are some related questions:
1. What is the cross product of two vectors?
2. How do you find the angle between two vectors using the dot product?
3. How does the direction of a resultant vector change if one vector is rotated by 90 degrees?
4. What are the properties of vector addition in three-dimensional space?
5. How do you determine if two vectors are parallel or perpendicular?

**Tip:** When dealing with vectors, it's essential to visualize them in a coordinate system, as this helps in understanding their resultant directions and magnitudes better.

Explore how to find the magnitude and direction of vector addition (A + B) and the dot product (AB) with detailed step-by-step solutions. Understand the Pythagorean theorem, dot product formula, and tangent function application. Suitable for Grades 10-12 students and anyone interested in vector operations.

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