A reflected laser and a sensor-distance check

Laser (Light Amplification by Stimulated Emission of Radiation) has many applications including medicine, data storage, military and industrial uses. It has the property of spatial coherence, which allows the laser beam to stay narrow over long distances. When a laser beam is projected onto a mirror at an angle, it reflects off the mirror at the same angle.

An engineer is designing a device that does industrial cutting using a laser beam. To make the device compact, the device has a mirror to reflect the beam before it leaves the device. The laser beam source is located at the origin OO. It projects an incident beam with direction vector i+2j+k\mathbf i+2\mathbf j+\mathbf k. The beam hits the mirror at the point PP with angle θ\theta. The mirror has an equation −x+2y+3z=12-x+2y+3z=12.

Q20 incident and reflected beams equal mirror anglesmirrorOPO′Incident beamReflected beamθθ==
Question diagram 2
Q20 incident and reflected beams equal mirror anglesmirrorOPO′Incident beamReflected beamθθ==

Native source reflection diagram

(i) Find the acute angle θ\theta that the beam makes with the mirror. marks

(ii) By finding O′O', the image of OO in the mirror, find a vector equation of the line that the reflected beam is on. marks

The engineer plans to install a sensor at (3,1,0)(3,1,0) to monitor the heat produced by the laser. For the sensor to work properly, the sensor must be less than 2 units away from either the incident or the reflected beam. Determine if the sensor will work properly. marks

[2017/YJC/Prelim/I/12]

Total marks: 13

Solution

(i)

Angle between a line and a plane

The beam has direction (1,2,1)(1,2,1); the mirror is −x+2y+3z=12-x+2y+3z=12. Find the acute angle θ\theta the beam makes with the mirror.

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Hint

  1. Angle with the normal, then the complement.

Answer · (i)

40.9∘40.9^\circ.

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The beam meets the mirror

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(ii)

Reflected ray

Find O′O', the image of OO in the mirror, and a vector equation of the reflected beam.

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Hint

  1. Foot NN of the perpendicular from OO: ON→=λ(−1,2,3)\overrightarrow{ON}=\lambda(-1,2,3).

  2. O′=2NO'=2N.

  3. The reflected beam is the line O′PO'P.

Answer · (ii)

O′=(−12,24,36)/7O\prime=(-12,24,36)/7; reflected ray r=(2,4,2)+t(13,2,−11)r=(2,4,2)+t(13,2,-11), t≥0t\ge0.

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The image method

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Sensor check

Distance from a point to a line

A sensor at B(3,1,0)B(3,1,0) must be less than 2 units from the incident or the reflected beam. Does it work?

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Hint

  1. Distance =∣OB→×d∣∣d∣=\frac{|\overrightarrow{OB}\times\mathbf d|}{|\mathbf d|} for each beam.

Answer · Sensor check

Does not work: incident distance 35/6\sqrt{35/6} and reflected distance 3275/294\sqrt{3275/294} both exceed 22.

Both perpendicular feet lie on the actual beams.

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The sensor and the two beams

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