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: Identify the given velocities
The velocity of the object is given as:
\[ \mathbf{v_0} = 2\hat{i} - 3\hat{j} + 4\hat{k} \]
The velocity of the mirror is given as:
\[ \mathbf{v_m} = \hat{i} - 4\hat{j} + 2\hat{k} \]

### Step 2: Determine the effect of the moving mirror
Since the normal to the mirror is along the x-axis, the x-component of the velocity of the image will be affected by the motion of the mirror. The y and z components will remain unchanged.

### Step 3: Calculate the x-component of the image's velocity
The formula for the x-component of the velocity of the image (\(v_{image_x}\)) is given by:
\[ v_{image_x} = 2 \cdot v_{m_x} - v_{0_x} \]
Where:
- \(v_{m_x}\) is the x-component of the mirror's velocity.
- \(v_{0_x}\) is the x-component of the object's velocity.

From the given velocities:
- \(v_{m_x} = 1\) (from \(\hat{i}\))
- \(v_{0_x} = 2\) (from \(2\hat{i}\))

Substituting these values into the formula:
\[ v_{image_x} = 2 \cdot 1 - 2 = 2 - 2 = 0 \]

### Step 4: Determine the y and z components of the image's velocity
Since the y and z components of the image's velocity remain the same as those of the object, we have:
- \(v_{image_y} = v_{0_y} = -3\)
- \(v_{image_z} = v_{0_z} = 4\)

### Step 5: Combine the components to find the velocity vector of the image
The velocity vector of the image can be expressed as:
\[ \mathbf{v_{image}} = v_{image_x} \hat{i} + v_{image_y} \hat{j} + v_{image_z} \hat{k} \]
Substituting the values we found:
\[ \mathbf{v_{image}} = 0\hat{i} - 3\hat{j} + 4\hat{k} \]
Thus, the final answer is:
\[ \mathbf{v_{image}} = -3\hat{j} + 4\hat{k} \]

### Final Answer
The velocity vector of the image is:
\[ \mathbf{v_{image}} = -3\hat{j} + 4\hat{k} \]

---