Class 12 Three Dimensional Geometry · Case Study4 Marksline_applications
Class 12 Maths · Three Dimensional Geometry · Case Study #93
Two roads are modelled by L₁:r=t(1,2,−1) and L₂:r=(3,3,0)+s(2,1,1). These are spatial line parameters, not synchronized travel times. Answer (i),(ii), and either (iii)(a) or (iii)(b).
Part (1) 1 Marks
Find Cartesian equations of L₁.
Answer: x=y/2=−z.
- Equate coordinates: x=t,y=2t,z=−t.
- Eliminating t gives x=y/2=−z.
Part (2) 1 Marks
Find direction cosines of L₂ in its stated orientation.
Answer: (2,1,1)/√6.
- The direction (2,1,1) has norm √6.
- Divide each component by √6.
Part (3) 2 Marks
Find the shortest distance between the roads.
Answer: 0.
- Solve t(1,2,−1)=(3,3,0)+s(2,1,1).
- The coordinates give t=1,s=−1, and both points are (1,2,−1).
- The lines intersect, so their shortest distance is 0.
Part (4) 2 Marks
Find the geometric meeting point of the two roads.
Answer: (1,2,−1).
- Equate x,y,z of the two line parametrizations.
- t=3+2s, 2t=3+s, −t=s give t=1,s=−1.
- The common point is (1,2,−1); no collision time follows without travel data.