Step-by-Step Deconstruction
Topic: Lines in 3D Space & Shortest Distance
1. Extract Data from Given Equations
Standard form: (x – x₁) / a = (y – y₁) / b = (z – z₁) / c
- Line A (l₁): Point a₁ = (2, -1, 3), Direction vector b₁ = 3î – 2ĵ + 4k̂
- Line B (l₂): Point a₂ = (1, 3, -2), Direction vector b₂ = 2î + 1ĵ – 3k̂
(i) Check if Parallel
Lines are parallel if their direction ratios are proportional: a₁/a₂ = b₁/b₂ = c₁/c₂.
Check ratios: 3/2 ≠ -2/1 ≠ 4/-3.
Conclusion: The direction ratios are not proportional, so the tracks are not parallel.
(ii) Equation of Solar Panel Line
The line is parallel to l₁, so it has the same direction vector: b = 3î – 2ĵ + 4k̂.
It passes through the point (1, -2, -3).
Equation: (x – 1) / 3 = (y – (-2)) / -2 = (z – (-3)) / 4
Final Equation: (x – 1) / 3 = (y + 2) / -2 = (z + 3) / 4
(iii) (a) Equation of Pedestrian Pathway
The pathway is perpendicular to both l₁ and l₂. Its direction d is the cross product b₁ × b₂.
d = | î ĵ k̂ |
| 3 -2 4 |
| 2 1 -3 |
= î(6 – 4) – ĵ(-9 – 8) + k̂(3 – (-4))
= 2î + 17ĵ + 7k̂
It passes through point (3, 2, 1).
Final Equation: (x – 3) / 2 = (y – 2) / 17 = (z – 1) / 7
(iii) (b) Shortest Distance between l₁ and l₂
Formula for skew lines: d = | (a₂ – a₁) · (b₁ × b₂) | / | b₁ × b₂ |
- a₂ – a₁ = (1-2)î + (3-(-1))ĵ + (-2-3)k̂ = -î + 4ĵ – 5k̂
- b₁ × b₂ = 2î + 17ĵ + 7k̂ (from part a)
- Dot Product: (-1)(2) + (4)(17) + (-5)(7) = -2 + 68 – 35 = 31
- Magnitude of b₁ × b₂: √(2² + 17² + 7²) = √(4 + 289 + 49) = √342
Shortest Distance d = 31 / √342 units
⚠️ Silly Mistake Audit: The Cross-Product Trap
Calculating the cross product b₁ × b₂ is the most common source of error here, especially with the negative sign for the ĵ component. A single sign mistake will make your direction vector for part (iii)(a) and your shortest distance calculation for part (iii)(b) completely wrong. Always double-check your determinant expansion!