A point object infront of a plane mirror has velocity `-2 hat i+ 3 hat j + 4 hat k` and the mirror has velocity `3 hat i- 2 hat j+ 4 hat k`. The x-axis is along the normal of the pane mirrro. Find the velocity of the image.

Answer

To find the velocity of the image of a point object in front of a plane mirror, we can follow these steps:

### Step 1: Identify the velocities of the object and the mirror
The velocity of the object \( \vec{v}_o \) is given as:
\[ \vec{v}_o = -2 \hat{i} + 3 \hat{j} + 4 \hat{k} \]
The velocity of the mirror \( \vec{v}_m \) is given as:
\[ \vec{v}_m = 3 \hat{i} - 2 \hat{j} + 4 \hat{k} \]

### Step 2: Determine the velocity of the image with respect to the object
The velocity of the image \( \vec{v}_i \) with respect to the object is calculated by reversing the x-component of the object's velocity (since the x-axis is along the normal to the mirror). Thus:
\[ \vec{v}_{i/o} = -\vec{v}_o = 2 \hat{i} - 3 \hat{j} - 4 \hat{k} \]

### Step 3: Calculate the velocity of the image with respect to the mirror
The velocity of the image with respect to the mirror is given by:
\[ \vec{v}_{i/m} = 2 \vec{v}_m \]
Calculating this gives:
\[ \vec{v}_{i/m} = 2(3 \hat{i} - 2 \hat{j} + 4 \hat{k}) = 6 \hat{i} - 4 \hat{j} + 8 \hat{k} \]

### Step 4: Combine the two components to find the total velocity of the image
The total velocity of the image \( \vec{v}_i \) is the sum of the velocity of the image with respect to the object and the velocity of the image with respect to the mirror:
\[ \vec{v}_i = \vec{v}_{i/o} + \vec{v}_{i/m} \]
Substituting the values:
\[ \vec{v}_i = (2 \hat{i} + 3 \hat{j} + 4 \hat{k}) + (6 \hat{i} - 4 \hat{j} + 8 \hat{k}) \]
Combining the components:
\[ \vec{v}_i = (2 + 6) \hat{i} + (3 - 4) \hat{j} + (4 + 8) \hat{k} = 8 \hat{i} - 1 \hat{j} + 12 \hat{k} \]

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