If `a,b,` and `c` are the unit vectors along the incident ray, reflected ray and the outward normal to the reflector. Then

Options

A. `b=a=c`
B. `b=a+(a.c)c`
C. `b=2a-c`
D. `b=a-2(a.c)c`

Answer

To solve the problem, we need to analyze the relationship between the unit vectors \( \mathbf{a} \), \( \mathbf{b} \), and \( \mathbf{c} \) which represent the incident ray, reflected ray, and the outward normal to the reflector, respectively.

### Step-by-Step Solution:

1. **Define the Vectors**:
- Let \( \mathbf{a} \) be the unit vector along the incident ray.
- Let \( \mathbf{b} \) be the unit vector along the reflected ray.
- Let \( \mathbf{c} \) be the unit vector along the outward normal to the reflector.

2. **Understand the Geometry**:
- When a ray of light strikes a reflective surface, it makes an angle \( \alpha \) with the normal (which is represented by vector \( \mathbf{c} \)).
- The angle of reflection is equal to the angle of incidence, so the reflected ray also makes an angle \( \alpha \) with the normal.

3. **Express the Vectors in Components**:
- The incident vector \( \mathbf{a} \) can be expressed in terms of its components:
\[ \mathbf{a} = \sin(\alpha) \hat{i} - \cos(\alpha) \hat{j} \]
- The reflected vector \( \mathbf{b} \) can be expressed similarly:
\[ \mathbf{b} = \sin(\alpha) \hat{i} + \cos(\alpha) \hat{j} \]

4. **Relate the Vectors**:
- The relationship between the incident vector \( \mathbf{a} \) and the reflected vector \( \mathbf{b} \) can be derived from the law of reflection:
\[ \mathbf{b} = \mathbf{a} - 2(\mathbf{a} \cdot \mathbf{c}) \mathbf{c} \]
- Here, \( \mathbf{a} \cdot \mathbf{c} \) gives the projection of \( \mathbf{a} \) onto the normal vector \( \mathbf{c} \).

5. **Calculate the Dot Product**:
- The dot product \( \mathbf{a} \cdot \mathbf{c} \) can be calculated as:
\[ \mathbf{a} \cdot \mathbf{c} = |\mathbf{a}| |\mathbf{c}| \cos(180^\circ - \alpha) = -\cos(\alpha) \]
- Since both \( \mathbf{a} \) and \( \mathbf{c} \) are unit vectors, their magnitudes are 1.

6. **Substitute Back**:
- Substitute the value of \( \mathbf{a} \cdot \mathbf{c} \) back into the equation for \( \mathbf{b} \):
\[ \mathbf{b} = \mathbf{a} - 2(-\cos(\alpha)) \mathbf{c} \]
\[ \mathbf{b} = \mathbf{a} + 2\cos(\alpha) \mathbf{c} \]

7. **Final Relationship**:
- Thus, we have established the relationship between the reflected ray \( \mathbf{b} \) and the normal \( \mathbf{c} \):
\[ \mathbf{b} = \mathbf{a} + 2(\mathbf{a} \cdot \mathbf{c}) \mathbf{c} \]

### Conclusion:
The relationship between the unit vectors along the incident ray, reflected ray, and the outward normal to the reflector can be expressed as:
\[ \mathbf{b} = \mathbf{a} + 2(\mathbf{a} \cdot \mathbf{c}) \mathbf{c} \]