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

Given:


A = (1,1,1)


B = (2,-1,1)


C = (1,2,-3)

Find the distance between A and the line that passes through points B and C?
Calculate the vectors BA and BC.





u = BA = %26lt;A - B%26gt; = %26lt;1-2, 1+1, 1-1%26gt; = %26lt;-1, 2, 0%26gt;


v = BC = %26lt;C - B%26gt; = %26lt;1-2, 2+1, -3-1%26gt; = %26lt;-1, 3, -4%26gt;





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 = %26lt;-1, 2, 0%26gt; X %26lt;-1, 3, -4%26gt; = %26lt;-8, -4, -1%26gt;





|| 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