`hatn_(1)` is the unit vector along incident ray, `hatn_(2)` along normal and `hatn_(3)` is the unit vector along reflected ray, then which of the following must be true?

Options

A. `hatn_(1)*hatn_(2)=0`
B. `hatn_(1)*hatn_(3)=0`
C. `(hatn_(1)xxhatn_(2))*hatn_(3)=0`
D. `(hatn_(1)xxhatn_(2))xxhatn_(3)=0`

Answer

To solve the problem, we need to analyze the relationships between the unit vectors along the incident ray, the normal, and the reflected ray in the context of reflection.

### Step-by-Step Solution:

1. **Define the Unit Vectors**:
- Let \( \hat{n}_1 \) be the unit vector along the incident ray.
- Let \( \hat{n}_2 \) be the unit vector along the normal to the surface.
- Let \( \hat{n}_3 \) be the unit vector along the reflected ray.

2. **Understanding Reflection**:
- According to the law of reflection, the angle of incidence (the angle between the incident ray and the normal) is equal to the angle of reflection (the angle between the reflected ray and the normal).

3. **Coplanarity of Vectors**:
- The incident ray, the reflected ray, and the normal all lie in the same plane. This means that the three vectors \( \hat{n}_1 \), \( \hat{n}_2 \), and \( \hat{n}_3 \) are coplanar.

4. **Using the Scalar Triple Product**:
- For three vectors to be coplanar, the scalar triple product must be zero. Thus, we can write:
\[ \hat{n}_1 \cdot (\hat{n}_2 \times \hat{n}_3) = 0 \]
- This indicates that \( \hat{n}_1 \) is perpendicular to the vector \( \hat{n}_2 \times \hat{n}_3 \).

5. **Cross Product of Unit Vectors**:
- The cross product \( \hat{n}_2 \times \hat{n}_3 \) gives a vector that is perpendicular to both \( \hat{n}_2 \) and \( \hat{n}_3 \). Since \( \hat{n}_1 \) is also in the same plane formed by \( \hat{n}_2 \) and \( \hat{n}_3 \), the dot product being zero confirms that they are indeed coplanar.

6. **Conclusion**:
- Therefore, the relationship that must hold true is:
\[ \hat{n}_1 \cdot (\hat{n}_2 \times \hat{n}_3) = 0 \]
- This means that the scalar triple product of the three unit vectors is zero, confirming their coplanarity.

### Final Answer:
The correct statement that must be true is:
\[ \hat{n}_1 \cdot (\hat{n}_2 \times \hat{n}_3) = 0 \]