1. Idea of reflection in vector form

When a ray with unit direction vector $\vec a$ is reflected from a surface whose outward unit normal is $\vec c$, the component of $\vec a$ parallel to the surface remains unchanged, while the component along the normal reverses sign.

2. Resolve the incident direction into normal and tangential parts

The component of $\vec a$ along $\vec c$ is

So the tangential component is

3. Apply reflection rule

After reflection:

• tangential component stays same,
• normal component changes sign.

Hence reflected vector $\vec b$ is

Therefore,

4. Match with options

• A: $\vec b = \vec a + 2\vec c$
  Incorrect, since it ignores the projection factor $(\vec a \cdot \vec c)$.

• B: $\vec b = \vec a - 2(\vec a \cdot \vec c)\vec c$
  Correct.

• C: $\vec b = 2\vec a + \vec c$
  Incorrect.

• D: $\vec b = \vec a - \vec c$
  Incorrect.

5. Final answer

The correct relation is

So the correct option is B.