A ray of light travelling in the direction `(1)/(2)``(hati,+sqrt3hatj)` is incident on a plane mirror. After reflection, it travels along the direction `(1)/(2)(hati-sqrt3hatj)` . The angle of incidence is

Options

A. `30^(@)`
B. `45^(@)`
C. `60^(@)`
D. `75^(@)`

Explanation

To find the angle of incidence when a ray of light strikes a plane mirror, we can follow these steps:

### Step 1: Identify the direction vectors
The incident ray is given by the direction vector:
\[ \vec{I} = \frac{1}{2} \hat{i} + \sqrt{3} \hat{j} \]
The reflected ray is given by the direction vector:
\[ \vec{R} = \frac{1}{2} \hat{i} - \sqrt{3} \hat{j} \]

### Step 2: Understand the normal to the mirror
Since the only component that changes upon reflection is the y-component, we can assume that the normal to the plane mirror is along the y-axis (i.e., in the direction of \(\hat{j}\)). Therefore, the normal vector can be represented as:
\[ \vec{N} = \hat{j} \]

### Step 3: Calculate the angle of incidence
The angle of incidence (\(i\)) is defined as the angle between the incident ray vector and the normal vector. We can find this angle using the dot product formula:
\[ \cos(i) = \frac{\vec{I} \cdot \vec{N}}{|\vec{I}| |\vec{N}|} \]

### Step 4: Calculate the dot product \(\vec{I} \cdot \vec{N}\)
Calculating the dot product:
\[ \vec{I} \cdot \vec{N} = \left(\frac{1}{2} \hat{i} + \sqrt{3} \hat{j}\right) \cdot \hat{j} = 0 + \sqrt{3} = \sqrt{3} \]

### Step 5: Calculate the magnitudes
Next, we need to find the magnitude of \(\vec{I}\) and \(\vec{N}\):
\[ |\vec{I}| = \sqrt{\left(\frac{1}{2}\right)^2 + (\sqrt{3})^2} = \sqrt{\frac{1}{4} + 3} = \sqrt{\frac{13}{4}} = \frac{\sqrt{13}}{2} \]
\[ |\vec{N}| = |\hat{j}| = 1 \]

### Step 6: Substitute into the cosine formula
Now substituting into the cosine formula:
\[ \cos(i) = \frac{\sqrt{3}}{\frac{\sqrt{13}}{2} \cdot 1} = \frac{2\sqrt{3}}{\sqrt{13}} \]

### Step 7: Find the angle of incidence
To find the angle \(i\), we take the inverse cosine:
\[ i = \cos^{-1}\left(\frac{2\sqrt{3}}{\sqrt{13}}\right) \]

### Step 8: Calculate the angle
Using a calculator or trigonometric tables, we can find the angle:
\[ i \approx 30^\circ \]

Thus, the angle of incidence is:
\[ \boxed{30^\circ} \]

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