A plane mirror is moving with velocity `4 (hat i) + 5 (hat j) + 8 (hat k)`. A point object in front of the mirror moves with a velocity `3 (hat i) + 4 (hat j) + 5 (hat k)`. Here, `hat k` is along the normal to the plane mirror and facing towards the object. The velocity of the image is

Options

A. `-3 (hat i) - 4 (hat j) + 5 (hat k)`
B. `3 (hat i) + 4 (hat j) + 11 (hat k)`
C. `-4 (hat i) + 5 (hat j) + 11 (hat k)`
D. `7 (hat i) +9 (hat j) + 3 (hat k)`

Explanation

To find the velocity of the image formed by a moving plane mirror, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Velocities**:
- Velocity of the mirror, \( \vec{V_m} = 4\hat{i} + 5\hat{j} + 8\hat{k} \)
- Velocity of the object, \( \vec{V_o} = 3\hat{i} + 4\hat{j} + 5\hat{k} \)

2. **Calculate the Velocity of the Object with Respect to the Mirror**:
- The velocity of the object with respect to the mirror is given by:
\[ \vec{V_{om}} = \vec{V_o} - \vec{V_m} \]
- Substituting the values:
\[ \vec{V_{om}} = (3\hat{i} + 4\hat{j} + 5\hat{k}) - (4\hat{i} + 5\hat{j} + 8\hat{k}) \]
- Performing the subtraction:
\[ \vec{V_{om}} = (3 - 4)\hat{i} + (4 - 5)\hat{j} + (5 - 8)\hat{k} = -1\hat{i} - 1\hat{j} - 3\hat{k} \]

3. **Determine the Velocity of the Image with Respect to the Mirror**:
- The velocity of the image with respect to the mirror is the negative of the velocity of the object with respect to the mirror:
\[ \vec{V_{im}} = -\vec{V_{om}} = -(-1\hat{i} - 1\hat{j} - 3\hat{k}) = 1\hat{i} + 1\hat{j} + 3\hat{k} \]

4. **Calculate the Velocity of the Image**:
- The velocity of the image is given by:
\[ \vec{V_i} = \vec{V_{im}} + \vec{V_m} \]
- Substituting the values:
\[ \vec{V_i} = (1\hat{i} + 1\hat{j} + 3\hat{k}) + (4\hat{i} + 5\hat{j} + 8\hat{k}) \]
- Performing the addition:
\[ \vec{V_i} = (1 + 4)\hat{i} + (1 + 5)\hat{j} + (3 + 8)\hat{k} = 5\hat{i} + 6\hat{j} + 11\hat{k} \]

5. **Final Result**:
- The velocity of the image is:
\[ \vec{V_i} = 5\hat{i} + 6\hat{j} + 11\hat{k} \]