How Do You Prove a Vector is Perpendicular to a Plane?

[iampaul]

Homework Statement

The vector r=xi+yj+zk, called the position vector, points from the origin(0,0,0) to an arbitrary point in space with coordinates(x,y,z).

Homework Equations

Use what you know about vectors to prove the following: All points (x,y,z) that satisfy the equation Ax+By+Cz=0 where A,B and C are constants,lie in a plane that passes through the origin and that is, perpendicular to the vector n=Ai+Bj+Ck.

The Attempt at a Solution

I can only show that the position vector pointing at (x,y,z) is perpendicular to n=Ai+Bj+Ck.. I did this by using cosa= (Ax+By+Cz)/rn to find the angle between n and r. Ax+By+Cz=0 so cosa=0 and a=90 degrees. But i can't still show that n is perpendicular to the plane. I'm also confused if the problem also asks to show that Ax+By+Cz=0 is a plane.Can somebody please help me. Thanks

 

[rbnvrw]

Well, the plane is defined as \vec{r} \cdot \vec{n} = 0. This states that r is perpendicular to n. So every point r for which this equation holds, lies in the plane. See also this page on planes: http://mathworld.wolfram.com/Plane.html. Hope this helps!