A boat stems in a river with velocity ` 2hati + hatj` with respecitve to the ground. The river water flows with a velocity ` -3hati - 4hatj` with respect to the ground. What is the relative velocity of boat w.r.t river water ?

Explanation

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

### Step 1: Identify the given velocities
- Velocity of the boat with respect to the ground, \( \vec{V}_{BR} = 2\hat{i} + \hat{j} \)
- Velocity of the river with respect to the ground, \( \vec{V}_{RG} = -3\hat{i} - 4\hat{j} \)

### Step 2: Use the formula for relative velocity
The relative velocity of the boat with respect to the river can be calculated using the formula:
\[ \vec{V}_{BR} = \vec{V}_{BG} - \vec{V}_{RG} \]
where \( \vec{V}_{BG} \) is the velocity of the boat with respect to the ground and \( \vec{V}_{RG} \) is the velocity of the river with respect to the ground.

### Step 3: Substitute the values into the formula
Substituting the given values into the formula:
\[ \vec{V}_{BR} = (2\hat{i} + \hat{j}) - (-3\hat{i} - 4\hat{j}) \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ \vec{V}_{BR} = (2\hat{i} + \hat{j}) + (3\hat{i} + 4\hat{j}) \]
Combine the like terms:
\[ \vec{V}_{BR} = (2 + 3)\hat{i} + (1 + 4)\hat{j} = 5\hat{i} + 5\hat{j} \]

### Step 5: Write the final answer
Thus, the relative velocity of the boat with respect to the river is:
\[ \vec{V}_{BR} = 5\hat{i} + 5\hat{j} \]
or, it can also be expressed as:
\[ \vec{V}_{BR} = 5(\hat{i} + \hat{j}) \]

### Final Answer:
The relative velocity of the boat with respect to the river water is \( 5\hat{i} + 5\hat{j} \).

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