We choose the unit vectors perpendicular to the mirror as the `x, y, z` axes in space. Then after reflection from the mirror with normal along `x` axis
`overset rarr(e) = overset rrarr(e) - 2 hati (hati. overset rarr(e)) = oversetrarr(e)_(x)hati = e_(y) hatj + e_(z) hatk`
where `hati, hatj, htak` are the basic unit vectors. after a second reflection from the `2nd` mirror say along `y` axis.
`oversetrarr(e)= oversetrarr(e)-2hatj (hatj.overset rarr(e)) = -e_(x)hati-e_(y) hath+e_(z) hatk`
Finally after the third reflection
`oversetrarr(e)=-e_(x)hati-e_(y)hatj -e_(z) hatk =-oversetrarr(e)`
