Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Say a plane P has a given equation ax+by+cz=d. Given a point $(x_0, y_0, z_0)$ that is not included in P. When $(x_0, y_0, z_0)$ is plugged into $f(x,y,z)=ax+by+cz-d$ and it outputs some nonzero $e$.

What does $e$ represent?

What information does this give us about $(x_0, y_0, z_0)$ in relation to P?

I believe that the set of all points such that $f(x,y,z)=e$ form a plane with equation $ax+by+cz=d+e$. Is this all that's useful about the extra information? The corresponding parallel plane that the point $(x_0, y_0, z_0)$ belongs to?

share|cite|improve this question
up vote 2 down vote accepted

You are right that $ax+by+cd=d+e$ describes another plane. More precisely a plane parallel to the first one.

The value of $e$ is related to the distance between these two planes (or between the point and the plane in the first part of the question). More precisely, this distance equals $\frac{|e|}{\sqrt{ a^2+b^2+c^2}}$. The sign of $e$ determines the side of the given plane. With this point of view, the points in the given plane are simply those points having distance $0$ from the plane :)

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.