A force of `7hati + 6hatk` newton makes a body move on a rough plane with a velocity of `3 hatj +4 hatk ms^(-1)`. Calculate the power in watt.

Options

A. 24
B. 34
C. 21
D. 45

Answer

To calculate the power in watts, we need to use the formula for power, which is the dot product of the force vector and the velocity vector.

### Step-by-Step Solution:

1. **Identify the Force and Velocity Vectors:**
- The force vector \( \vec{F} \) is given as:
\[ \vec{F} = 7\hat{i} + 6\hat{k} \text{ N} \]
- The velocity vector \( \vec{v} \) is given as:
\[ \vec{v} = 3\hat{j} + 4\hat{k} \text{ m/s} \]

2. **Write the Force and Velocity in Component Form:**
- We can express the vectors in a more detailed component form:
\[ \vec{F} = 7\hat{i} + 0\hat{j} + 6\hat{k} \]
\[ \vec{v} = 0\hat{i} + 3\hat{j} + 4\hat{k} \]

3. **Calculate the Dot Product \( \vec{F} \cdot \vec{v} \):**
- The dot product is calculated as follows:
\[ \vec{F} \cdot \vec{v} = (7)(0) + (0)(3) + (6)(4) \]
- Simplifying this gives:
\[ \vec{F} \cdot \vec{v} = 0 + 0 + 24 = 24 \text{ W} \]

4. **Conclusion:**
- The power \( P \) is equal to the dot product of the force and velocity:
\[ P = 24 \text{ W} \]

### Final Answer:
The power is \( 24 \) watts.

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