If the incident ray on a surface is along the unit vector `vec v`, the reflected ray is along the unit vector `vec w` and the normal is along the unit vector ` vec a` outwards, express `vec w` in terms of `vec a` and `vec v`

Answer

`hat(w) = hat(v) -2(hat(a)"."hat(v)) hat(a)`

Explanation

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)`
`:. , hat(w) -hat(v) = lambda hat(a)`

On squaring both sides we get
`rArr 1+1 - hat(w) , hat(v) = lambda^(2) rArr 2-2 cos 2theta = lambda^(2)`
`rArr lambda = 2 sin theta`
where `2 theta ` is the angle between `hat(v) " and " hat(w)`
Hence `hat(w) - hat(v) =2 sin theta . hat(a) =2 cos (90^(@) - theta) hat(a) =-(2 hat(a) "." hat(v)) hat(a)`
`rArr hat(w) = hat(v) -2 (hat(a) ". " hat(v)) hat(a)`