If incident ray, refracted ray and normal ray are represented by unit vectors `vec a, vec b ,and vec c` then relation between them is ?

Options

A. `vec a - vec b = vec c`
B. `vec a . (vec b xxvec c) =0`
C. `vec a + vec c = 2vec b`
D. `vec a xx (vec b xxvec c) =0`

Answer

To find the relation between the incident ray, refracted ray, and the normal ray represented by unit vectors \(\vec{a}\), \(\vec{b}\), and \(\vec{c}\), we can analyze their geometric and vector properties.

### Step-by-Step Solution:

1. **Understanding the Vectors**:
- Let \(\vec{a}\) represent the incident ray.
- Let \(\vec{b}\) represent the refracted ray.
- Let \(\vec{c}\) represent the normal to the surface at the point of incidence.

2. **Using the Cross Product**:
- The cross product \(\vec{b} \times \vec{c}\) gives a vector that is perpendicular to the plane formed by \(\vec{b}\) and \(\vec{c}\).
- This means that \(\vec{a}\) (the incident ray) must be in the same plane as \(\vec{b}\) and \(\vec{c}\).

3. **Dot Product Condition**:
- The condition for \(\vec{a}\) being in the plane formed by \(\vec{b}\) and \(\vec{c}\) can be expressed mathematically as:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]
- This equation states that the dot product of \(\vec{a}\) with the vector perpendicular to the plane (i.e., \(\vec{b} \times \vec{c}\)) is zero, indicating that \(\vec{a}\) lies in the plane.

4. **Analyzing the Other Relations**:
- The relation \(\vec{a} + \vec{c} = 2\vec{b}\) is not generally true in the context of refraction and reflection.
- The relation \(\vec{a} \times (\vec{b} \times \vec{c}) = 0\) is also not valid since \(\vec{b} \times \vec{c}\) is not necessarily zero.

5. **Conclusion**:
- The only valid relation among the given options is:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]

### Final Answer:
The relation between the incident ray, refracted ray, and normal ray is:
\[ \vec{a} \cdot (\vec{b} \times \vec{c}) = 0 \]