`vec(A) and vec(B)` are two vectors such that `vec(A)` is 2 m long and is `60^(@)` above the x - axis in the first quadrant `vec(B)` is 2 m long and is `60^(@)` below the x - axis in the fourth quadrant .
Draw diagram to show `vec(a) + vec(B) , vec(A) - vec(B) and vec(B) -vec(A)`
(b) What is the magnitude and direction of `vec(A) + vec(B) , vec(A) - vec(B) and vec(B) - vec(A)`

Explanation

To solve the problem, we will follow these steps:

### Step 1: Draw the Vectors
1. **Vector A**: Draw a vector of length 2 m at an angle of 60 degrees above the x-axis in the first quadrant.
2. **Vector B**: Draw a vector of length 2 m at an angle of 60 degrees below the x-axis in the fourth quadrant.

### Step 2: Calculate the Components of the Vectors
- For **Vector A**:
- \( A_x = A \cos(60^\circ) = 2 \cos(60^\circ) = 2 \times \frac{1}{2} = 1 \, \text{m} \)
- \( A_y = A \sin(60^\circ) = 2 \sin(60^\circ) = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3} \, \text{m} \)

- For **Vector B**:
- \( B_x = B \cos(60^\circ) = 2 \cos(60^\circ) = 2 \times \frac{1}{2} = 1 \, \text{m} \)
- \( B_y = B \sin(-60^\circ) = 2 \sin(-60^\circ) = 2 \times -\frac{\sqrt{3}}{2} = -\sqrt{3} \, \text{m} \)

### Step 3: Calculate the Resultant Vectors
(a) **Vector A + Vector B**:
- \( R_{A+B} = (A_x + B_x, A_y + B_y) = (1 + 1, \sqrt{3} - \sqrt{3}) = (2, 0) \)
- Magnitude:
\[ |R_{A+B}| = \sqrt{(2)^2 + (0)^2} = 2 \, \text{m} \]
- Direction: Along the x-axis (0 degrees).

(b) **Vector A - Vector B**:
- \( R_{A-B} = (A_x - B_x, A_y - B_y) = (1 - 1, \sqrt{3} - (-\sqrt{3})) = (0, 2\sqrt{3}) \)
- Magnitude:
\[ |R_{A-B}| = \sqrt{(0)^2 + (2\sqrt{3})^2} = 2\sqrt{3} \approx 3.46 \, \text{m} \]
- Direction: 90 degrees (along the positive y-axis).

(c) **Vector B - Vector A**:
- \( R_{B-A} = (B_x - A_x, B_y - A_y) = (1 - 1, -\sqrt{3} - \sqrt{3}) = (0, -2\sqrt{3}) \)
- Magnitude:
\[ |R_{B-A}| = \sqrt{(0)^2 + (-2\sqrt{3})^2} = 2\sqrt{3} \approx 3.46 \, \text{m} \]
- Direction: 270 degrees (along the negative y-axis).

### Summary of Results
- **Magnitude and Direction of A + B**:
- Magnitude: 2 m
- Direction: 0 degrees (along the x-axis)

- **Magnitude and Direction of A - B**:
- Magnitude: \( \approx 3.46 \, \text{m} \)
- Direction: 90 degrees (along the positive y-axis)

- **Magnitude and Direction of B - A**:
- Magnitude: \( \approx 3.46 \, \text{m} \)
- Direction: 270 degrees (along the negative y-axis)