How to find min distance between line that doesn't intersect a plane

Question

Good morning,

How would you find the Min distance between a line and a plane (if the line does NOT intersect the plane)?

E.g.

Q(s) = (1,2,3) + s(-2,1,-1)

Plane: 3a + 5b - c = -3

Thanks for the help!

Answer 1

Since $$(-2,1,-1)(3,5,-1)=0,$$ we see that the line is parallel to the plane.

Just take a point in the line (the point $(1,2,3)$ for example) and find a distance between this point and the plane.

I got $$\frac{13}{\sqrt{35}}.$$

Answer 2

You have a formula for the (min) distance of point $M(x_0,y_0,z_0)$ to the plane $(\Pi)$ with equation $ux+vy+wz+t=0$: $$d(M,\Pi)=\frac{|ux_0+vy_0+wz_0+t|}{\sqrt{u^2+v^2+w^2\,}}.$$ If the line is parallel to the plane, this distance will be the same for all points of the line.