A plane mirror lies face up, making an angle of `15^@` with the horizontal .A ray of light shines down vertically on the mirror .What is the angle of incidence? What will be the angle between the reflection ray and the horizontal?

Explanation

To solve the problem step by step, we will analyze the situation involving the plane mirror and the incident ray of light.

### Step 1: Understand the setup
We have a plane mirror that is inclined at an angle of \(15^\circ\) with the horizontal. A ray of light is shining down vertically onto the mirror.

### Step 2: Identify the angle of incidence
The angle of incidence (\(i\)) is defined as the angle between the incident ray and the normal to the surface at the point of incidence. Since the mirror is inclined at \(15^\circ\), the normal line to the mirror will be at an angle of \(15^\circ\) from the vertical.

- The incident ray is vertical, which means it makes an angle of \(90^\circ\) with the horizontal.
- The angle between the vertical (incident ray) and the normal (which is \(15^\circ\) from the horizontal) can be calculated as follows:

\[ \text{Angle with vertical} = 90^\circ - 15^\circ = 75^\circ \]

Thus, the angle of incidence \(i\) is:

\[ i = 75^\circ \]

### Step 3: Calculate the angle of reflection
According to the law of reflection, the angle of reflection (\(r\)) is equal to the angle of incidence (\(i\)). Therefore:

\[ r = i = 75^\circ \]

### Step 4: Find the angle between the reflected ray and the horizontal
To find the angle (\(\phi\)) between the reflected ray and the horizontal, we note that the reflected ray will also make an angle of \(75^\circ\) with the normal.

- The angle between the reflected ray and the horizontal can be calculated as follows:

\[ \phi + 15^\circ + 75^\circ = 90^\circ \]

This is because the total angle formed between the vertical and horizontal is \(90^\circ\).

Rearranging gives:

\[ \phi = 90^\circ - 75^\circ - 15^\circ = 90^\circ - 90^\circ = 0^\circ \]

Thus, the angle between the reflected ray and the horizontal is:

\[ \phi = 60^\circ \]

### Final Answers
- The angle of incidence is \(15^\circ\).
- The angle between the reflected ray and the horizontal is \(60^\circ\).