A particle moves with a velocity of `(4hati-2hatj+2hatk)ms^(-1)` under the influence of a constant force `vecF=(10hati+3hatj-2hatk)N`. The instantaneous power applied to the particle is

Answer

To find the instantaneous power applied to the particle, we can use the formula for instantaneous power, which is given by the dot product of the force vector and the velocity vector.

### Step-by-Step Solution:

1. **Identify the Given Vectors**:
- The velocity vector \( \vec{V} \) is given as:
\[ \vec{V} = 4 \hat{i} - 2 \hat{j} + 2 \hat{k} \, \text{m/s} \]
- The force vector \( \vec{F} \) is given as:
\[ \vec{F} = 10 \hat{i} + 3 \hat{j} - 2 \hat{k} \, \text{N} \]

2. **Write the Formula for Instantaneous Power**:
- The instantaneous power \( P \) is given by:
\[ P = \vec{F} \cdot \vec{V} \]

3. **Calculate the Dot Product**:
- The dot product \( \vec{F} \cdot \vec{V} \) can be calculated as follows:
\[ \vec{F} \cdot \vec{V} = (10 \hat{i} + 3 \hat{j} - 2 \hat{k}) \cdot (4 \hat{i} - 2 \hat{j} + 2 \hat{k}) \]
- Using the properties of the dot product:
\[ = (10 \cdot 4) + (3 \cdot -2) + (-2 \cdot 2) \]
- Calculate each term:
- \( 10 \cdot 4 = 40 \)
- \( 3 \cdot -2 = -6 \)
- \( -2 \cdot 2 = -4 \)

4. **Combine the Results**:
- Now, combine all the results:
\[ P = 40 - 6 - 4 = 30 \, \text{W} \]

5. **Final Answer**:
- The instantaneous power applied to the particle is:
\[ P = 30 \, \text{W} \]