In Multivariable calculus, it seems planes are defined by a point and a vector normal to the plane. That makes sense, but whereas a single vector clearly isn't enough to define a plane, aren't two non-colinear vectors enough?
What I'm thinking is that since we need the cross-product of two vectors to find our normal vector in the first place, why not just use those two vectors to define our plane. After all, don't two non-colinear vectors define a basis in R2?
Remember, vectors don't have starting or ending positions, just directions. So take the vectors <1,0,0> and <0,1,0>. These vectors will define a plane that only goes in the x-y direction, but the problem is, they will work for any z-coordinate. So you need some starting point to anchor your plane.