Question

Incident ray is along the unit vector $\hat{v}$ and the reflected ray is along the unit vector $\hat{w}$. The normal is along unit vector $\hat{a}$ outwards. Express $\hat{w}$ in terms of $\hat{a}$ and $\hat{v}$.

Answer 1

Since, $\hat{v}$ is unit vector along the incident ray and $\hat{w}$ is the unit vector along the reflected ray.
Hence, $\hat{a}$ is a unit vector along the external bisector of $\hat{v}$ and ${\hat{w}}_i$
$\therefore \hat{w}-\hat{v}=\lambda \hat{a}$

On squaring both sides, we get
$\Rightarrow 1+1-\hat{w}\cdot \hat{v}={\lambda }^2\Rightarrow 2-2\cos 2\theta ={\lambda }^2$
$\Rightarrow \lambda =2\sin \theta$
where, $2\theta$ is the angle between $\hat{v}$ and $\hat{w}$.
Hence, $\hat{w}-\hat{v}=2\sin \theta \cdot \hat{a}=2\cos \left(90^{\circ }-\theta \right)\hat{a}=-(2\hat{a}\cdot \hat{v})\hat{a}$
$\Rightarrow \hat{w}=\hat{v}-2(\hat{a}\cdot \hat{v})\hat{a}$