Vectors and 10 points?

Question

Find the distance from the point (-2,1,5) to the straight line that joins the point (1,2,-5) and (7,5,-9).

Please show me in detail, plus the working so I can answer my other questions based on this.

Answer 1

Here's one way to do this.

Let P=(1,2,-5), Q = (7,5,-9), R = (-2,1,5)

Vector PQ = <6, 3, -4>

Parametric equations for the line through P and Q are

x = 1 + 6t

y = 2 + 3t

z = -5 - 4t

Let T be the point on the line such that line segment RT is perpendicular to the line. Then vector RT is perpendicular to vector PQ, so the dot product RT⋅PQ = 0:

RT = <3+6t, 1+3t, -10-4t>

RT⋅PQ = 0 ⇒ (3+6t)(6) + (1+3t)(3) + (-10-4t)(-4) = 0

⇒ 61 + 61t = 0

⇒ t = -1

so

RT = <-3, -2, -6>

The magnitude of RT is the distance between R and the line through P and Q. |RT| = √(9 + 4 + 36) = 7