Porisma

Worked solution

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.

Expand diagram

Use the scrollbars or arrow keys to inspect details.

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

Answer

θ=40.9°; OO′=127(−1,2,3)OO′=\frac{12}{7}(-1,2,3); reflected line r=(2,4,2)+γ(13,2,−11)r=(2,4,2)+\gamma(13,2,-11); sensor will not work, since both distances exceed 2.

(i) Use a line-plane sine

Normal n=(−1,2,3)\mathbf n=(-1,2,3) and incident direction (1,2,1)(1,2,1) give sin⁡θ=∣(1,2,1)⋅(−1,2,3)∣/(614)=6/84\sin\theta=|(1,2,1)\cdot(-1,2,3)|/(\sqrt6\sqrt{14})=6/\sqrt{84}. Therefore θ=40.9°.

How did this part go?

(ii) Reflect the source through its plane foot

Let F be the foot from O. Set OF=λ(−1,2,3)OF=\lambda(-1,2,3); since F lies in the mirror,14λ=1214\lambda=12, so λ=6/7. Hence OO′=2OF=127(−1,2,3)OO′=2OF=\frac{12}{7}(-1,2,3), and the position vector of O′ is that same vector because O is the origin.

How did this part go?

(ii) Find the incidence point P

OP=μ(1,2,1)OP=\mu(1,2,1). The mirror condition gives μ(1,2,1)⋅(−1,2,3)=12\mu(1,2,1)\cdot(-1,2,3)=12, so 6μ=12, μ=2 and OP=(2,4,2)OP=(2,4,2).

How did this part go?

(ii) Parameterise the reflected supporting line

O′P=OP−OO′=27(13,2,−11)O′P=OP-OO′=\frac27(13,2,-11). Thus the reflected line is r=(2,4,2)+γ(13,2,−11)\mathbf r=(2,4,2)+\gamma(13,2,-11), γ real. The outgoing beam uses γ≥0.

How did this part go?

(iii) Distance to the incident path

Let B=(3,1,0). The supporting-line distance is ∣(3,1,0)×(1,2,1)∣/6=∣(1,−3,5)∣/6=35/6>2|(3,1,0)\times(1,2,1)|/\sqrt6=|(1,-3,5)|/\sqrt6=\sqrt{35/6}>2. The perpendicular foot parameter 5/6 lies within the physical incident segment 0≤μ≤2, so this is also the beam-segment distance.

How did this part go?

(iii) Distance to the reflected path and conclusion

PB=(1,−3,−2)PB=(1,-3,-2). Its cross product with (13,2,−11)(13,2,-11) is (37,−15,41)(37,-15,41), so the distance is 3275/294=3275/294>2\sqrt{3275}/\sqrt{294}=\sqrt{3275/294}>2. The perpendicular foot parameter 29/294 is nonnegative, so it lies on the outgoing ray. Neither beam is less than 2 units away; the sensor will not work properly.

How did this part go?

Common mistakes

Check that a supporting-line distance applies to the actual beam segment or outgoing ray. Both foot parameters are valid here.

Solution notes

All 3 asks and source calculations retained with the original optical diagram reconstructed. This question has no printed marks; none invented. Q15 heading on PDF 34 retained in its native transcript alongside PDF 35–37. Additional foot-parameter checks justify using line distances for the actual beams.

Read the hint