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 $\hat{a}$ coutwards. Express $\hat{w}$ in terms of $\hat{a}$ and $\hat{v}$.

Answer 1

$\hat{v}$ 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 $\hat{w}$.

Hence, $\hat{w}$ - $\hat{v}$ $=\lambda$

$\hat{a}$ $\Rightarrow 1+1-$ $\hat{w}$ . $\hat{v}$ $={\lambda }^2$ or 2 2 cos 2 = $={\lambda }^2$ or $\lambda$ 2 sin $\theta$ where 2 $\theta$ is the angle between $\hat{v}$ and $\hat{w}$ Hence

$\hat{w}$ - $\hat{v}$ =2sin $\theta$ $\hat{a}$= 2cos $({90}^0-\theta )$

$\hat{a}$ = - $(2\hat{a}.\hat{v})\hat{a}\Rightarrow \hat{w}=\hat{v}-2\left(\widehat{a.\hat{v}}\right)\hat{a}.$