A plane mirror is moving with velocity ` -2 hati - 3 hatj + 4hatk `. A point object in front of the mirror move with a velocity `-3 hati + 4 hatj -4hatk `.
There ` hat j ` is along the normal to the plane mirror and facing towards the object. Find velocity of image.

Options

A. ` - 3 hati - 10 hatj - 4 hatk `
B. ` - 3 hatj + 11 hatj -4 hatk `
C. ` - 2 hatj + 11 hatj + 4hatk `
D. ` - 2 hatj - 10 hatj - 4 hatk `

Explanation

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

### Step 1: Identify the velocities
Given:
- Velocity of the mirror, \( \vec{V_m} = -2 \hat{i} - 2 \hat{j} + 4 \hat{k} \)
- Velocity of the object, \( \vec{V_o} = -3 \hat{i} + 4 \hat{j} - 4 \hat{k} \)

### Step 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:
\[ \vec{V_{o/m}} = \vec{V_o} - \vec{V_m} \]
Substituting the values:
\[ \vec{V_{o/m}} = (-3 \hat{i} + 4 \hat{j} - 4 \hat{k}) - (-2 \hat{i} - 2 \hat{j} + 4 \hat{k}) \]
Calculating this:
\[ \vec{V_{o/m}} = (-3 + 2) \hat{i} + (4 + 2) \hat{j} + (-4 - 4) \hat{k} \]
\[ \vec{V_{o/m}} = -1 \hat{i} + 6 \hat{j} - 8 \hat{k} \]

### Step 3: Determine the velocity of the image with respect to the mirror
The velocity of the image with respect to the mirror is obtained by reversing the component of velocity along the normal (which is in the \( \hat{j} \) direction) and keeping the other components the same:
\[ \vec{V_{i/m}} = -\vec{V_{o/m}} = -(-1 \hat{i} + 6 \hat{j} - 8 \hat{k}) = 1 \hat{i} - 6 \hat{j} + 8 \hat{k} \]

### Step 4: Calculate the velocity of the image
The velocity of the image is given by:
\[ \vec{V_i} = \vec{V_{i/m}} + \vec{V_m} \]
Substituting the values:
\[ \vec{V_i} = (1 \hat{i} - 6 \hat{j} + 8 \hat{k}) + (-2 \hat{i} - 2 \hat{j} + 4 \hat{k}) \]
Calculating this:
\[ \vec{V_i} = (1 - 2) \hat{i} + (-6 - 2) \hat{j} + (8 + 4) \hat{k} \]
\[ \vec{V_i} = -1 \hat{i} - 8 \hat{j} + 12 \hat{k} \]

### Final Answer
The velocity of the image is:
\[ \vec{V_i} = -1 \hat{i} - 8 \hat{j} + 12 \hat{k} \]