A boat is moving with a velocity 2i + 3j with respect to ground. The water in the river is moving with a velocity -2i - 3j with respect to ground. The relative velocity of the boat with respect to water is

Options

A. 4j
B. `-4j+6j`
C. 4j+6j
D. 6j

Explanation

To find the relative velocity of the boat with respect to the water, we can use the following formula:

\[ \text{Velocity of boat with respect to water} = \text{Velocity of boat with respect to ground} - \text{Velocity of water with respect to ground} \]

### Step-by-Step Solution:

1. **Identify the velocities:**
- Velocity of the boat with respect to the ground, \( \vec{V}_{bg} = 2\hat{i} + 3\hat{j} \)
- Velocity of the water with respect to the ground, \( \vec{V}_{wg} = -2\hat{i} - 3\hat{j} \)

2. **Apply the formula for relative velocity:**
\[ \vec{V}_{bw} = \vec{V}_{bg} - \vec{V}_{wg} \]

3. **Substitute the values:**
\[ \vec{V}_{bw} = (2\hat{i} + 3\hat{j}) - (-2\hat{i} - 3\hat{j}) \]

4. **Simplify the expression:**
- Distributing the negative sign:
\[ \vec{V}_{bw} = 2\hat{i} + 3\hat{j} + 2\hat{i} + 3\hat{j} \]
- Combine like terms:
\[ \vec{V}_{bw} = (2 + 2)\hat{i} + (3 + 3)\hat{j} = 4\hat{i} + 6\hat{j} \]

5. **Final result:**
The relative velocity of the boat with respect to the water is:
\[ \vec{V}_{bw} = 4\hat{i} + 6\hat{j} \]