A plane mirror placed at the origin has `hat(i)` as the normal vector to its reflecting surface. The mirror beings to translate with a velocity `hat(i) + hat(j) + hat(k)`. At the same time an object which was initially at `hat(i) + hat(j)` starts moving with a velocity `(hat(i)+hat(j)) m//s` Now choose the correct options.

Options

A. Initial position of the image will be `-hat(i)+hat(j)`
B. The velocity of the image will be `hat(i)+hat(j)`
C. The velocity of the imahe relative to the object will be zero
D. The velocity of the image relative to the mirror will be `-hat(k)`

Answer

To solve the problem, we will analyze the motion of the mirror and the object, and then find the velocity of the image formed by the mirror.

### Step 1: Understand the given parameters
- The plane mirror is at the origin with a normal vector along \(\hat{i}\).
- The mirror translates with a velocity \(\vec{V}_{\text{mirror}} = \hat{i} + \hat{j} + \hat{k}\).
- The object starts at position \(\hat{i} + \hat{j}\) and moves with a velocity \(\vec{V}_{\text{object}} = \hat{i} + \hat{j}\).

### 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:
\[ \vec{V}_{\text{object, mirror}} = \vec{V}_{\text{object}} - \vec{V}_{\text{mirror}} \]
Substituting the values:
\[ \vec{V}_{\text{object, mirror}} = (\hat{i} + \hat{j}) - (\hat{i} + \hat{j} + \hat{k}) = \hat{i} + \hat{j} - \hat{i} - \hat{j} - \hat{k} = -\hat{k} \, \text{m/s} \]

### Step 3: Determine the velocity of the image with respect to the mirror
The velocity of the image with respect to the mirror is given by reversing the x-component of the velocity of the object (since the mirror reflects the object):
\[ \vec{V}_{\text{image, mirror}} = (-\hat{i} + \hat{j} + \hat{k}) \text{ (keeping y and z components the same)} \]
Thus, the velocity of the image with respect to the mirror is:
\[ \vec{V}_{\text{image, mirror}} = -\hat{k} \, \text{m/s} \]

### Step 4: Calculate the velocity of the image
The velocity of the image can be calculated by adding the velocity of the mirror to the velocity of the image with respect to the mirror:
\[ \vec{V}_{\text{image}} = \vec{V}_{\text{image, mirror}} + \vec{V}_{\text{mirror}} \]
Substituting the values:
\[ \vec{V}_{\text{image}} = (-\hat{k}) + (\hat{i} + \hat{j} + \hat{k}) = \hat{i} + \hat{j} \]

### Step 5: Determine the velocity of the image with respect to the object
Now, we find the velocity of the image with respect to the object:
\[ \vec{V}_{\text{image, object}} = \vec{V}_{\text{image}} - \vec{V}_{\text{object}} \]
Substituting the values:
\[ \vec{V}_{\text{image, object}} = (\hat{i} + \hat{j}) - (\hat{i} + \hat{j}) = \vec{0} \]

### Step 6: Position of the image
The position of the image can be determined by reflecting the position of the object across the mirror. The initial position of the object is \(\hat{i} + \hat{j}\), and the image position will be:
\[ \text{Position of image} = (-1, 1) = -\hat{i} + \hat{j} \]

### Summary of Results
- Velocity of the image: \(\hat{i} + \hat{j}\)
- Velocity of the image with respect to the object: \(0\)
- Position of the image: \(-\hat{i} + \hat{j}\)