RI Tutorial 2C Vectors III Q5

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 \(O\). It projects an incident beam with direction vector \(\mathbf{i}+2\mathbf{j}+\mathbf{k}\). The beam hits the mirror at the point \(P\) with angle \(\theta \). The mirror has an equation \(-x+2y+3z=12\).

1. Find the acute angle \(\theta \) that the beam makes with the mirror.[2]
2. By finding \(O\,'\), the image of \(O\) in the mirror, find a vector equation of the line that the reflected beam is on.[7]
3. The engineer plans to install a sensor at \(\left( 3,\,\,1,\,\,0 \right)\) 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.[4]