A force `3hat i+4 hat j - 5 hatk` N is acting on a particle. If the velocity of particle at an instant is `2hati + 2 hat j + 2 hatk` m/s , find the instantaneous power developed by the force (in watts).

Explanation

To find the instantaneous power developed by the force acting on a particle, we will use the formula for power, which is given by 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 \( \mathbf{F} \) is given as:
\[ \mathbf{F} = 3 \hat{i} + 4 \hat{j} - 5 \hat{k} \, \text{N} \]
- The velocity vector \( \mathbf{v} \) is given as:
\[ \mathbf{v} = 2 \hat{i} + 2 \hat{j} + 2 \hat{k} \, \text{m/s} \]

2. **Use the Power Formula:**
- The instantaneous power \( P \) is calculated using the dot product:
\[ P = \mathbf{F} \cdot \mathbf{v} \]

3. **Calculate the Dot Product:**
- The dot product is calculated as follows:
\[ P = (3 \hat{i} + 4 \hat{j} - 5 \hat{k}) \cdot (2 \hat{i} + 2 \hat{j} + 2 \hat{k}) \]
- Expanding this, we use the properties of dot products:
\[ P = (3 \cdot 2)(\hat{i} \cdot \hat{i}) + (4 \cdot 2)(\hat{j} \cdot \hat{j}) + (-5 \cdot 2)(\hat{k} \cdot \hat{k}) \]
- Since \( \hat{i} \cdot \hat{i} = 1 \), \( \hat{j} \cdot \hat{j} = 1 \), and \( \hat{k} \cdot \hat{k} = 1 \), we have:
\[ P = 3 \cdot 2 + 4 \cdot 2 - 5 \cdot 2 \]

4. **Perform the Calculations:**
- Calculate each term:
\[ P = 6 + 8 - 10 \]
- Combine the results:
\[ P = 6 + 8 - 10 = 4 \, \text{watts} \]

5. **Final Answer:**
- The instantaneous power developed by the force is:
\[ P = 4 \, \text{watts} \]