IIT-JEE 2005Question 1
If the incident ray on a surface is along the unit vector $\hat v$, the reflected ray is along the unit vector $\hat w$ and the normal is along the unit vector $\hat a$ outwards, express $\hat w$ in terms of $\hat a$ and $\hat v$. [figure: mirror surface with outward normal $\hat a$, incident ray $\hat v$ arriving and reflected ray $\hat w$ leaving, symmetric about the normal]
Vector Algebra
Answer$\hat w=\hat v-2(\hat a\cdot\hat v)\hat a$
Solution
On reflection the component of $\hat v$ along the normal reverses and the tangential component is unchanged: $\hat w=\hat v-2(\hat a\cdot\hat v)\hat a$.