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

`hatw = hatv - 2( hata.hatv)hata`

Explanation

Given that the incident ray is along `hatv` . The reflected ray is along `hata` .outwards, The given figure can be redrawn as shown in fig. A .2.30
We know that he icident , ray the refilected ray , and the normal lie in a plan, and the angle of incidence is equal to the angle of reflection.
Therefore, `hata` will be along the angle bisecter of ` hatw and -hatv`, i.e.
` hata= (hatw + (-hatv))/(|hatw - hatv|)`
But `hata` is a unit vector
Where `|hatw- hatv| = OC= 2OP`
`=2 |hatw|theta= 2costheta`
substituting this value in (i), we get
` hata= (hatw - hatv)/(2 cos theta)`
` or hatw - hatv - 2 (hata - hatv) hata " " ( hata. hatv =- cos theta)`