A plane mirror is moving with velocity `4 (hat i) + 4 (hat j) + 8 (hat k)`. A point object in front of the mirror moves with a velocity `3 (hat i) + 4 (hat j) + 5 (hat k)`. Here, `hat k` is along the normal to the plane mirror and facing towards the object. The velocity of the image is

Options

A. `-3 (hat i) - 4 (hat j) + 5 (hat k)`
B. `3 (hat i) + 4 (hat j) + 11 (hat k)`
C. `-4 (hat i) + 5 (hat j) + 11 (hat k)`
D. `7 (hat i) +9 (hat j) + 3 (hat k)`

Explanation

To find the velocity of the image in the given scenario, we can follow these steps:

### Step 1: Identify the given velocities
- The velocity of the mirror, \( \vec{V_m} \) is given as:
\[ \vec{V_m} = 4 \hat{i} + 4 \hat{j} + 8 \hat{k} \]
- The velocity of the object, \( \vec{V_o} \) is given as:
\[ \vec{V_o} = 3 \hat{i} + 4 \hat{j} + 5 \hat{k} \]

### Step 2: Calculate the relative velocity of the object with respect to the mirror
The relative velocity of the object with respect to the mirror, \( \vec{V_{om}} \), can be calculated using the formula:
\[ \vec{V_{om}} = \vec{V_o} - \vec{V_m} \]

Substituting the values:
\[ \vec{V_{om}} = (3 \hat{i} + 4 \hat{j} + 5 \hat{k}) - (4 \hat{i} + 4 \hat{j} + 8 \hat{k}) \]

Calculating each component:
- For \( \hat{i} \): \( 3 - 4 = -1 \)
- For \( \hat{j} \): \( 4 - 4 = 0 \)
- For \( \hat{k} \): \( 5 - 8 = -3 \)

Thus, we have:
\[ \vec{V_{om}} = -1 \hat{i} + 0 \hat{j} - 3 \hat{k} = -\hat{i} - 3 \hat{k} \]

### Step 3: Find the velocity of the image with respect to the mirror
The velocity of the image with respect to the mirror, \( \vec{V_{im}} \), is related to the velocity of the object with respect to the mirror by the equation:
\[ \vec{V_{im}} = -\vec{V_{om}} \]
This means:
\[ \vec{V_{im}} = -(-\hat{i} - 3 \hat{k}) = \hat{i} + 3 \hat{k} \]

### Step 4: Calculate the velocity of the image
Now, we can find the velocity of the image, \( \vec{V_i} \), using the formula:
\[ \vec{V_i} = \vec{V_{im}} + \vec{V_m} \]

Substituting the values:
\[ \vec{V_i} = (\hat{i} + 3 \hat{k}) + (4 \hat{i} + 4 \hat{j} + 8 \hat{k}) \]

Calculating each component:
- For \( \hat{i} \): \( 1 + 4 = 5 \)
- For \( \hat{j} \): \( 0 + 4 = 4 \)
- For \( \hat{k} \): \( 3 + 8 = 11 \)

Thus, we have:
\[ \vec{V_i} = 5 \hat{i} + 4 \hat{j} + 11 \hat{k} \]

### Final Answer
The velocity of the image is:
\[ \vec{V_i} = 5 \hat{i} + 4 \hat{j} + 11 \hat{k} \]

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