Find the distance between A and the line that passes through points B and C?

Question

Given:
A = (1,1,1)
B = (2,-1,1)
C = (1,2,-3)

Answer 1

Calculate the vectors BA and BC.

u = BA = = <1-2, 1+1, 1-1> = <-1, 2, 0>
v = BC = = <1-2, 2+1, -3-1> = <-1, 3, -4>

Let
d = distance between point A and line BC
θ = angle between vectors u and v

Calculate:

|| v || = √[(-1)² + 3² + (-4)²] = √(1 + 9 + 16) = √26

u X v = <-1, 2, 0> X <-1, 3, -4> = <-8, -4, -1>

|| u X v || = √[(-8)² + (-4)² + (-1)²] = √(64 + 16 + 1) = √81 = 9

We have:
sinθ = d / || u ||

We also have:
|| u X v || = || u || || v || sinθ = || u || || v || (d / || u ||) = || v || d

d = || u X v || / || v || = 9/√26 ≈ 1.7650452