A mirror and a source of light are situated at the origin `O` and at a point on `O X ,` respectively. A ray of light from the source strikes the mirror and is reflected. If the direction ratios of the normal to the plane are `1,-1,1,` then find the `D C s` of the reflected ray.

Explanation

Let the source of light be situated at `A(a, 0, 0)`, where `ane0`.
Let `OA` be the incident ray and `OB` the reflected ray.
`ON` is the normal to the mirror at O. Therefore,
`" "angleAON=angleNOB=theta//2" ""(say)"`
Direction ratios of `OA` are `a`, 0 and 0 and so its direction cosines are 1, 0 and 0.
Direction ratios of `ON` are `1//sqrt(3)`, `-1//sqrt(3)` and `1//sqrt(3)`. Therefore,
`" "angleAON=angleNOB=(theta//2)" ""(say)"`
`" "cos(theta//2)=1//sqrt(3)`
Let `l, m and n` be the direction cosines of the reflected ray OB. Then
`" "(l+1)/(2cos(theta//2))=(1)/(sqrt(3)),(m+0)/(2cos(theta//2))=-(1)/(sqrt(3)) and (n+0)/(2cos(theta//2))=(1)/(sqrt(3))`
`rArr" "l=(2)/(3)-1, m=(-2)/(3), n=(2)/(3)`
`or" "l=-(1)/(3), m=-(2)/(3), n=(2)/(3)`