Let L1 be the line passing through the points Q13 1 5 and Q2

Let L1 be the line passing through the points Q1=(3, 1, 5) and Q2=(1, 5, 3) and let L2 be the line passing through the point P1=(6, 10, 2) with direction vector d=[3, 1, 2]T. Determine whether L1 and L2 intersect. If so, find the point of intersection Q.

Solution

The line L1 passes through the point ( -3,1,5) and has the direction vector ( -2, 6, 2)S . The line L2 passes through the point ( -6, -10, -2) and has the direction vector ( 3,1,2)T . Then, the parametric equations of the lines L1 and L2 are as under:

Line L1:

x = - 3 - 2S; y = 1 + 6S; z = 5 + 2S

Line L2:

x = - 6 + 3T ; y = -10+ T ; z = -2 + 2T

The lines L1 and L2 intersect if we can find values for S and T such that:

-3 – 2S = - 6 + 3T

1 + 6S = -10 + T

5 + 2S = -2 + 2T

These equations have the solution S = -3/2 and T = 2.

Now, on substituting S = -3/2 in the equation of the line L1, we get x = 0, y = -8, z = 2

On substituting T = 2 in the equation of the line L2, we get x = 0, y = - 8 and z = 2.

Thus the point ( 0, - 8, 2) is the point of intersection of the lines L1 and L2.