`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 question regarding the relationships between the unit vectors of the incident ray (\(\hat{n}_1\)), normal (\(\hat{n}_2\)), and reflected ray (\(\hat{n}_3\)), we will analyze each option step by step.

### Step-by-Step Solution:

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

2. **Angle Relationships**:
- According to the law of reflection, the angle of incidence (\(\theta\)) is equal to the angle of reflection.
- Therefore, the angle between \(\hat{n}_1\) and \(\hat{n}_2\) is \(\theta\) and the angle between \(\hat{n}_3\) and \(\hat{n}_2\) is also \(\theta\).

3. **Analyzing the Options**:
- **Option A**: \(\hat{n}_1 \cdot \hat{n}_2\)
- The dot product \(\hat{n}_1 \cdot \hat{n}_2 = |\hat{n}_1||\hat{n}_2|\cos(\theta)\).
- Since \(\hat{n}_1\) and \(\hat{n}_2\) are unit vectors, this simplifies to \(\cos(\theta)\).
- This is not necessarily zero unless \(\theta = 90^\circ\), which is not the case here. Thus, this option is **false**.

- **Option B**: \(\hat{n}_1 \cdot \hat{n}_3\)
- The angle between \(\hat{n}_1\) and \(\hat{n}_3\) is \(180^\circ - 2\theta\).
- Therefore, \(\hat{n}_1 \cdot \hat{n}_3 = \cos(180^\circ - 2\theta) = -\cos(2\theta)\).
- This is also not necessarily zero unless \(\theta = 90^\circ\). Thus, this option is **false**.

- **Option C**: \((\hat{n}_1 \times \hat{n}_2) \cdot \hat{n}_3\)
- The cross product \(\hat{n}_1 \times \hat{n}_2\) gives a vector perpendicular to both \(\hat{n}_1\) and \(\hat{n}_2\).
- Since \(\hat{n}_3\) lies in the plane formed by \(\hat{n}_1\) and \(\hat{n}_2\), the dot product will be zero.
- Therefore, this option is **true**.

- **Option D**: \((\hat{n}_1 \times \hat{n}_2) \times \hat{n}_3\)
- This expression represents a vector that is not guaranteed to be zero since it involves a cross product.
- Thus, this option is **false**.

4. **Conclusion**:
- The only true statement is from **Option C**: \((\hat{n}_1 \times \hat{n}_2) \cdot \hat{n}_3 = 0\).