Case Study Challenge
3D Geometry in Metro Rail Design
An engineer is designing a new metro rail network in a city. Initially, two metro lines, Line A and Line B, each consisting of multiple stations are designed.
*Image for representational purpose only.
The track for Line A is represented by the line l₁:
(x – 2) / 3 = (y + 1) / -2 = (z – 3) / 4
While the track for Line B is represented by the line l₂:
(x – 1) / 2 = (y – 3) / 1 = (z + 2) / -3
Based on the above information, answer the following questions:
| (i) | Find whether the two metro tracks are parallel. | [1] |
| (ii) | Solar panels are to be installed on the rooftop of the metro stations. Determine the equation of the line representing the placement of solar panels on the rooftop of Line A’s stations, given that panels are to be positioned parallel to Line A’s track (l₁) and pass through the point (1, – 2, – 3). | [1] |
| (iii) | (a) To connect the stations, a pedestrian pathway perpendicular to the two metro lines is to be constructed which passes through point (3, 2, 1). Determine the equation of the pedestrian walkway.
— OR —
(b) Find the shortest distance between Line A and Line B. |
[2] |