Suppose that three coordinate planes are mirrored and a light ray given by the v

Question

Suppose that three coordinate planes are mirrored and a light ray given by the vector a = ( al,a2,a3 ) first strikes the xz - plane, then the reflected ray strikes the xy - plane and the reflected ray of this ray again strikes the yz - plane . Use the fact that the angle of incidence equals the angle of reflection to show that the direction of the reflected ray is given by b = ( al, - a2,a3 ) . Deduce that, after being reflected by all three mutually perpendicular mirrors, the resulting ray is parallel to the initial ray.

Explanation / Answer

To start off with, let's consider a two dimensional vector (a1,a2). If the slope were m and it were reflected off the x-axis, the new slope would be -m. Think about it - if the slope is 0, the new slope is still 0 (and this is -0). If the slope is straight down (-infinity), the new slope is straight up (+infinity). This means the vector goes from (a1,a2) to (a1,-a2). Applying this to three dimensional space, to reflect (a1,a2,a3) off a plane would negate the term that is perpendicular to the plane. Thus, when (a1,a2,a3) is reflected off the xz-plane, y is negated and the reflected vector is (a1,-a2,a3). When (a1,-a2,a3) is reflected off the yz-plane, x is negated and the vector is now (-a1,-a2,a3). It can be seen to make this vector a multiple of the original, we just need to reflect it off of the xy-plane and get (-a1,-a2,-a3). This vector is the same as -(a1,a2,a3), and that is a multiple of the first vector. I can upload a picture in two dimensions if necessary, or you can draw one yourself of a line being reflected off the x-axis.

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