Question

Let L1 be the line passing through the points Q1=(-5, 1,-4) and Q2=(1,-8,-1) and let L2 be the line passing through the point P1=(-10, 16,-5) with direction vector d=[-1,-1,-1]^T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q

Answer 1

Q 3. The direction vectors of the line L₁ is (1+5, -8-1, -1+4)^T i.e. ( 6, -9, 3)^T . The parametric equation of direction vectors of the line L₁ is (x,y,z) = (-5,1,-4)+t( 6, -9, 3) i.e. = -5+t, y = 1-9t, z = -4+3t.

The parametric equation of the line L₂ is (x,y,z) = (-10,16,-5)+s( -1,-1,-1) i.e. x = -10-s, y = 16-s , z = -5-s.

The lines L₁ and L₂ are not parallel as their direction vectors are not proportional. If these 2 lines were to intersect we would have three equations in two unknowns s and t as under:

-5+t= -10-s or, t+s = -5…(1), 1-9t = 16-s or, -9t+s = 15…(2) and -4+3t = -5-s or, 3t+s = -1…(3).

The augmentedmatrix of this linear system is A (say) =

1	1	-5
-9	1	15
3	1	-1

To solve the above linear system, we need to reduce A to its RREF which is

1	0	0
0	1	0
0	0	1

This implies that the above linear system is inconsistent as we cannot have 0 = 1.

Hence the lines L₁ and L₂ do not intersect.