Question

Find the shortest distance and the vector equation of the line of shortest distance between the lines given by

vector r = vector(2j - 3k) + λ vector(2i - j)

and vector r = vector(4i + 3k) + μ vector(3i + j + k) 

Answer 1

The equation of given line in cartesian form can be written as 

(x - 0)/2 = (y - 2)/-1 = (z + 2)/0 = λ ...(i)

and (x - 4)/3 = (y - 0)/1 = (z - 3)/1 μ ...(ii)

Line (i) passes through point (0, 2, -3) and d.r. is (2,-1,0). line (ii) passes through point (4,0,3) and d.r. is (3,1,1). From the given vector form, we can find variable point on each line, say

It vector PQ is taken as the shortest distance vector, then it should be perpendicular to both I₁, and I₂

Hence, shortest distance 

= PQ = √((2 - 1)² + (1 + 1)² + (-3 - 2)²)

= √(1² + 2² + 52) = √30

 Also, vector equation of the shortest distance PQ is 

vector r = (Position vector of P) + t.PQ 

= (2i + j - 3k) + t.(-i - 2j + 5k)