Question

Let L₁ be the line passing through the point P₁=(−1, −2, −4) with direction vector →d=[1, 1, 1]^T, and let L₂ be the line passing through the point P₂=(−2, 4, −1) with the same direction vector.
Find the shortest distance d between these two lines, and find a point Q₁ on L₁ and a point Q₂ on L₂ so that d(Q₁,Q₂) = d. Use the square root symbol '√' where needed to give an exact value for your answer.

Answer 1

Part i)

Line L1:

The point P1=<-1,-2,-4> with direction vector u=<1,1,1>

Line L2:

The point P2=<-2,4,-1> with direcion vector u=<1,1,1>

Since the direction vectors for L1 and L2 are same implies the lines L1 and L2 are parallel. The distance between parallel lines is

The line P1P2 is

The direction vector u is

Find the cross product of the vectors P1P2 and u

That is

The magntiude of the direction vector is

Therefore,

Part ii)

Let, Q1=P1=<-1,-2,-4>.

Any point on L2 is

Distance between the two points is

That is

Differentiate w.r.t t

That is

For minmum distance

Therefore, the point Q2 is

Distance between Q1 and Q2 is