# Point-Plane Distance?

1. Aug 26, 2010

Hey I just have a little concern about the distance between a plane & a point not on the plane.

In the picture here:
[PLAIN]http://img838.imageshack.us/img838/4197/normalyy.jpg [Broken]

we can define the equation of a line passing through B down to the plane and beyond as

X = B + tN

If we want to find the equation of the line passing through the point P in the plane
through to A we'll use the idea of the dot product due to it's orthogonal characteristic.

(X - A)•N = 0

I want to point out that X is going to have to be equal to the point P in the picture so

X = P = B + tN

and

(X - A)•N = (P - A)•N = (B + tN - A)•N = 0

We'll fineagle it to get

(B + tN - A)•N = 0

(B - A)•N + tNN = 0

tNN = - (B - A)•N

tNN = (A - B)•N

$$t \ = \ \frac{ ( \overline{A} \ - \ \overline{B}) \cdot \overline{N}}{ \overline{N} \cdot \overline{N}}$$

Then you can sub this into your original equation

X = B + tN

To find the actual value at the point P.

Using this couldn't you just find the norm of this numerical value to find the length of
the line connecting B to P?

Then you could go on to use the General Pythagoras Theorem and get

$$|| \overline{A} \ + \ \overline{B} || \ = \ || \overline{AP} || \ + \ || \overline{PB} ||$$

Last edited by a moderator: May 4, 2017