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Question 17

The incident ray, reflected ray and the outward drawn normal are denoted by the unit vectors $$\vec{a}$$, $$\vec{b}$$ and $$\vec{c}$$ respectively. Then choose the correct relation for these vectors.

Let $$\vec{a}$$ be the unit vector along the incident ray, $$\vec{b}$$ be the unit vector along the reflected ray, and $$\vec{c}$$ be the unit vector along the outward drawn normal to the reflecting surface.

By the law of reflection, the component of the incident ray along the surface (tangential component) remains unchanged, while the component along the normal reverses direction.

The component of $$\vec{a}$$ along the normal is $$(\vec{a} \cdot \vec{c})\vec{c}$$. Since the incident ray points toward the surface, $$\vec{a} \cdot \vec{c} < 0$$. The tangential component of $$\vec{a}$$ is $$\vec{a} - (\vec{a} \cdot \vec{c})\vec{c}$$.

For the reflected ray, the tangential component stays the same and the normal component reverses: $$\vec{b} = \vec{a} - (\vec{a} \cdot \vec{c})\vec{c} - (\vec{a} \cdot \vec{c})\vec{c} = \vec{a} - 2(\vec{a} \cdot \vec{c})\vec{c}$$.

This gives the standard reflection formula: $$\vec{b} = \vec{a} - 2(\vec{a} \cdot \vec{c})\vec{c}$$.

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