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Find a reflected laser line and check a sensor threshold

A laser source at O projects an incident beam with direction i+2j+k\mathbf i+2\mathbf j+\mathbf k. It hits a mirror at P with angle θ and reflects at the same angle. The mirror is −x+2y+3z=12-x+2y+3z=12.

(i) Find the acute angle θ the beam makes with the mirror.

(ii) By finding O′, the image of O in the mirror, find a vector equation of the line containing the reflected beam.

(iii) A sensor at (3,1,0)(3,1,0) must be less than 2 units from either the incident or reflected beam to work properly. Determine whether it will work.

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Mirror shown as a horizontal line through P. O lies above it, O′ the reflection below it; their common foot F is on the mirror and OF=FO′. Incident O→P and reflected P→upper-right beams make equal acute angles theta with the mirror. Dashed O′P extends the reflected line backward. Independent planar reconstruction of the source optical diagram, not a projection of its spatial coordinates.
Mirror shown as a horizontal line through P. O lies above it, O′ the reflection below it; their common foot F is on the mirror and OF=FO′. Incident O→P and reflected P→upper-right beams make equal acute angles theta with the mirror. Dashed O′P extends the reflected line backward. Independent planar reconstruction of the source optical diagram, not a projection of its spatial coordinates.
Mirror shown as a horizontal line through P. O lies above it, O′ the reflection below it; their common foot F is on the mirror and OF=FO′. Incident O→P and reflected P→upper-right beams make equal acute angles theta with the mirror. Dashed O′P extends the reflected line backward. Independent planar reconstruction of the source optical diagram, not a projection of its spatial coordinates.

[YJC Prelim 9758/2017/01/Q12]

CJC · Review problems · Question 15

Reflect the source through the plane, then join its image to the incidence point.

Read the worked solution