A unit vector along the incident ray of light is `hat(i)`. The unit vector for the corresponding refracted ray of light is `hat(r ).hat(n)`, a unit vector normal to the boundary of the medium and directed towards the incident medium. If `mu` is the refractive index of the medium, then snell's law (second law) of refraction is

Options

A. a. `hat(i)xxhat(n)=mu(hat(n)+vec(r))`
B. `hat(i).hat(n)=mu(hat(r ).hat(n))`
C. `hat(i)xxhat(n)=mu(hat(r )xxhat(n))`
D. `hat(i)xxhat(n)=mu(hat(r )xxhat(n))`

Explanation

To derive Snell's law in vector form based on the given information, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Unit Vectors**:
- Let the unit vector along the incident ray of light be represented as \(\hat{i}\).
- The unit vector normal to the boundary of the medium directed towards the incident medium is \(\hat{n}\).
- The unit vector for the refracted ray of light is represented as \(\hat{r}\).

2. **Understanding the Refractive Index**:
- The refractive index of the first medium (incident medium) is denoted as \(\mu_1\) and for the second medium (refracted medium) as \(\mu_2\).
- Given that \(\mu_1 = 1\) (for air) and \(\mu_2 = \mu\) (for the second medium), we can denote these values accordingly.

3. **Vector Form of Snell's Law**:
- Snell's law relates the angles of incidence and refraction to the refractive indices of the two media. In vector form, it can be expressed as:
\[ \mu_1 \cdot \hat{i} \times \hat{n} = \mu_2 \cdot \hat{r} \times \hat{n} \]

4. **Substituting the Refractive Indices**:
- Substituting the values of \(\mu_1\) and \(\mu_2\) into the equation, we get:
\[ 1 \cdot \hat{i} \times \hat{n} = \mu \cdot \hat{r} \times \hat{n} \]
- This simplifies to:
\[ \hat{i} \times \hat{n} = \mu \cdot \hat{r} \times \hat{n} \]

5. **Final Form of Snell's Law**:
- Rearranging the equation gives us the final vector form of Snell's law:
\[ \hat{i} \times \hat{n} = \mu \cdot \hat{r} \times \hat{n} \]

### Conclusion:
The vector form of Snell's law is given by:
\[ \hat{i} \times \hat{n} = \mu \cdot \hat{r} \times \hat{n} \]