Distance between Two Points

Question

Find the distance the point P(-7,2,3) is to the line through the two points Q(-2,-1,3) and R(-5,0,0).

If you could explain the steps you took to solve the problem, that would be very helpful.

I tried it and I got (sqrt(315)) / (sqrt(19)) which is wrong.

Answer 1

The line through Q(-2,-1,3) and R(-5,0,0) can be found as follows:

Vector v=QR is〈-5-(-2),0-(-1),0-3〉=〈-3,1,-3〉

So the line is parameterized by (x,y,z) = Q+vt = (-3t-5,t,-3t).

The squared distance of the general point on this line from $P$ is

s = (-7-(-3t-5))² + (2-t)² + (3-(-3t))²

= (3t-2)² + (2-t)² + (3+3t)²

= 9t² - 12t + 4 + 4 - 4t + t² + 9 + 18t + 9t²

= 19t² + 2t + 17

This is a quadratic so the minimum is at the parabola vertex:

t = -b/(2a) = -2/(2*19) = -1/19

Plug this in:

s(-1/19) = 19(-1/19)² + 2(-1/19) + 17 = 1/19 - 2/19 + 323/19 = 322/19

so the distance is √(322/19).