Calculate the distance between the lines

Question

Calculate the distance between the lines

L1: x=-5+6t, y=0+2t, z=4+4t

L2: x=-7+7s, y=-14+4s, z=14+7s

Distance: D=?

Answer 1

Direction vector of 1st line is a = <6, 2, 4>

Direction vector of 2nd line is b = < 7, 4, 7>

So normal vector of 2 lines is Ii j k I

n = a x b = I6 2 4I = - 2i - 14j + 10k

I7 4 7I

Point A(- 5, 0, 4) is point on L_1, point B(- 7, - 14, 14) is point on L₂.

So shortest distance D between L₁ and L₂: vector AB = <- 2, - 14, 10>;

D = Iproj_nABI = IAB•nI/InI = I4 + 196 + 100I/(4 + 196 + 100)^1/2 = 10√3 = 17.32