A point object is moving with velocity `v_(0) =2 hat(i) - 3 hat(j ) + 4hat( k)` in front of a moving plane mirror whose normal is along x - axis.The mirror is moving with velocity `v_(m) = hat(i) - 4 hat(j) + 2 hat( k ) `. Find the velocity vector of image

Options

A. `- 5 hat(j)`
B. `- 3 hat(j) + 4 hat(k)`
C. `-4 hat(j)+ 2 hat(k)`
D. `2 hat(i) - 3 hat(j) + 2 hat(k)`

Answer

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

### Step 1: Write down the velocities
The velocity of the object \( \vec{v_0} \) and the velocity of the mirror \( \vec{v_m} \) are given as follows:
- Velocity of the object:
\[ \vec{v_0} = 2\hat{i} - 3\hat{j} + 4\hat{k} \]
- Velocity of the mirror:
\[ \vec{v_m} = \hat{i} - 4\hat{j} + 2\hat{k} \]

### Step 2: Find the velocity of the object with respect to the mirror
To find the relative velocity of the object with respect to the mirror, we subtract the velocity of the mirror from the velocity of the object:
\[ \vec{v_{om}} = \vec{v_0} - \vec{v_m} \]
Substituting the values:
\[ \vec{v_{om}} = (2\hat{i} - 3\hat{j} + 4\hat{k}) - (\hat{i} - 4\hat{j} + 2\hat{k}) \]
Calculating this gives:
\[ \vec{v_{om}} = (2 - 1)\hat{i} + (-3 + 4)\hat{j} + (4 - 2)\hat{k} = 1\hat{i} + 1\hat{j} + 2\hat{k} \]

### Step 3: Determine the velocity of the image with respect to the mirror
In a plane mirror, the velocity of the image with respect to the mirror is the same as the velocity of the object with respect to the mirror, but the component along the normal (x-direction) is inverted. Since the normal of the mirror is along the x-axis, we change the sign of the x-component:
\[ \vec{v_{im}} = -\hat{i} + \hat{j} + 2\hat{k} \]

### Step 4: Find the velocity of the image
To find the velocity of the image, we add the velocity of the mirror to the velocity of the image with respect to the mirror:
\[ \vec{v_i} = \vec{v_{im}} + \vec{v_m} \]
Substituting the values:
\[ \vec{v_i} = (-\hat{i} + \hat{j} + 2\hat{k}) + (\hat{i} - 4\hat{j} + 2\hat{k}) \]
Calculating this gives:
\[ \vec{v_i} = (-1 + 1)\hat{i} + (1 - 4)\hat{j} + (2 + 2)\hat{k} = 0\hat{i} - 3\hat{j} + 4\hat{k} \]

### Final Answer
Thus, the velocity vector of the image is:
\[ \vec{v_i} = -3\hat{j} + 4\hat{k} \]