Distance between plane and a line?

Question

Cant figure this question out, can anyone help me out please?
Consider the plane:

0 x - 2 y + 6 z = 7 .

and the line:

[x,y,z]^T = [ -2,3,8 ] + s[ -5,4,-4]^T.

Find the smallest distance between the two.

The distance is

Answer 1

The smallest distance is zero.

Note if s=35/32, the point on the line is [-2-175/32,3+35/8,8-35/8] = [-238/32,59/8,29/8].

But note that this point ALSO lies on the plane:

-2*(59/8)+6*(29/8)=56/8=7

So the smallest distance between the two is zero.

Hope this helped!

Answer 2

If the line and plane are not parallel then the smallest distance between the two is zero. Let's check to see if they are parallel. If they are parallel then the normal vector to the plane will be perpendicular to the directional vector of the line. If this is true the dot product of the vectors will be zero.

The normal vector n, of the plane is:

n = <0, -2, 6>

The directional vector of the line r, is:

r = <-5, 4, -4>

Take the dot product.

n • r = <0, -2, 6> • <-5, 4, -4> = 0 - 8 - 24 = - 32 ≠ 0

Therefore the line and plane are not parallel and they intersect. The minimum distance between them therefore is zero.