Determine a vector equation for the plane that is parallel to the xy -plane and passes through the point (4, 1, 3).

Question

Determine a vector equation for the plane that is parallel to the xy -plane and passes through the point (4, 1, 3).

Answer 1

A normal to to xy-plane which will also be normal to any plane parallel to to xy-plane is <0,0,c> where c is a nonzero number.  A line in the plane parallel to the xy-plane is L = <x-4, y-1, z-3>.  Dotting the normal and line we have

 

<0,0,c>•<x-4, y-1, z-3> = 0

 

which is the vector equation of the plane.