A particle moves with a velocity `v=(5hati-3hatj + 6hatk) ms^(-1)` under the influence of a constant force
`F =(10hati + 10hatj + 20hath) N`, the instantaneous power applied to the particle is.

Options

A. `200 W`
B. `320 W`
C. `140 W`
D. `170 W`

Explanation

To find the instantaneous power applied to the particle, we can use the formula for power in terms of force and velocity, which is given by:

\[ P = \mathbf{F} \cdot \mathbf{v} \]

where \( P \) is the power, \( \mathbf{F} \) is the force vector, and \( \mathbf{v} \) is the velocity vector.

### Step-by-Step Solution:

1. **Identify the Force and Velocity Vectors:**
- Given the force vector:
\[ \mathbf{F} = 10 \hat{i} + 10 \hat{j} + 20 \hat{k} \, \text{N} \]
- Given the velocity vector:
\[ \mathbf{v} = 5 \hat{i} - 3 \hat{j} + 6 \hat{k} \, \text{m/s} \]

2. **Calculate the Dot Product:**
- The dot product \( \mathbf{F} \cdot \mathbf{v} \) is calculated as follows:
\[ \mathbf{F} \cdot \mathbf{v} = (10 \hat{i} + 10 \hat{j} + 20 \hat{k}) \cdot (5 \hat{i} - 3 \hat{j} + 6 \hat{k}) \]
- This expands to:
\[ = (10 \times 5) + (10 \times -3) + (20 \times 6) \]

3. **Perform the Multiplications:**
- Calculate each term:
\[ = 50 - 30 + 120 \]

4. **Combine the Results:**
- Now, add the results together:
\[ = 50 - 30 + 120 = 140 \]

5. **Final Result for Power:**
- The instantaneous power \( P \) is:
\[ P = 140 \, \text{W} \]

### Conclusion:
The instantaneous power applied to the particle is **140 watts**.