NJC Vectors II Tutorial Q13

A water jet cutter uses a mixture of water and abrasive material to cut metals and stones under high pressure. This is a preferred method than laser especially when the material to be cut is sensitive to high temperatures.

A triangular laminar is to be cut from a rectangular sheet of metal. The water jet cutter passes through three points with coordinates \(\left( 2,0,0 \right)\), \(\left( 4,0,0 \right)\) and \(\left( 3,6,1 \right)\) such that these three points form the vertices of the triangular laminar.

1. Find the equation of the triangular laminar in scalar product form.

To punch a hole through the triangular laminar, the nozzle of the water jet cutter is adjusted to the position \(\left( 3,4,\alpha \right)\) such that the perpendicular distance between the nozzle and the laminar is \(10\) units.

2. By finding the foot of perpendicular from the nozzle to the laminar, find the possible values of \(\alpha \).

One side of the triangular laminar is soldered at an angle to a square disc of negligible thickness. The equation of the square disc is \(z=0\).

3. Find the acute angle between the triangular laminar and the disc.
4. State the equation of the line of intersection between the laminar and the disc.

The square disc rests on a wood block with thickness \(3\) units.

5. Find the equation of the opposite surface of the wood block that is not in contact with the disc.