A boat is moving with a velocity `3 hat i+ 4hat j` with respect to ground. The water in the river is moving with a velocity `-3 hat i - 4 hat j` with respect to ground. The relative velocity of the boat with respect to water is.

Options

A. `8 hat j`
B. `-6 hat i - 8 hat j`
C. `6 hat i+ 8 hat j`
D. `5 sqrt(2))`

Explanation

To find the relative velocity of the boat with respect to the water, we can follow these steps:

### Step 1: Identify the given velocities
- The velocity of the boat with respect to the ground is given as:
\[ \vec{V}_{\text{boat}} = 3\hat{i} + 4\hat{j} \]
- The velocity of the water with respect to the ground is given as:
\[ \vec{V}_{\text{water}} = -3\hat{i} - 4\hat{j} \]

### Step 2: Write the formula for relative velocity
The relative velocity of the boat with respect to the water can be calculated using the formula:
\[ \vec{V}_{\text{boat, water}} = \vec{V}_{\text{boat}} - \vec{V}_{\text{water}} \]

### Step 3: Substitute the known values into the formula
Substituting the values we identified in Step 1 into the formula from Step 2:
\[ \vec{V}_{\text{boat, water}} = (3\hat{i} + 4\hat{j}) - (-3\hat{i} - 4\hat{j}) \]

### Step 4: Simplify the expression
When we subtract the velocity of the water, we need to change the signs:
\[ \vec{V}_{\text{boat, water}} = (3\hat{i} + 4\hat{j}) + (3\hat{i} + 4\hat{j}) \]
This simplifies to:
\[ \vec{V}_{\text{boat, water}} = 3\hat{i} + 4\hat{j} + 3\hat{i} + 4\hat{j} = (3 + 3)\hat{i} + (4 + 4)\hat{j} = 6\hat{i} + 8\hat{j} \]

### Step 5: State the final answer
Thus, the relative velocity of the boat with respect to the water is:
\[ \vec{V}_{\text{boat, water}} = 6\hat{i} + 8\hat{j} \]