Prove that if are not all zero, then the equation: represents a plane and is a normal vector to the plane.
Hint if a 0, then is equivalent to
My Calc III teacher loves proofs, but I'm not real good with them. How can I prove this? I'm not even sure how to approach it. Can anyone help? Thanks!
Well, the usual equation for a plane looks like this:
where is a unit vector normal to the surface, is a vector to the arbitrary point on the plane, and is a vector to any particular point on the plane.
Can you get your equation to look like mine? If so, you're done.
I know that , but is this enough to prove it? Where did you get the usual equation for a plane from? Thanks!
It is simply using the fact that two vectors are perpendicular if and only if their dot product is 0. Let be any fixed point and be any other point. Then the vector connecting them is [tex]<x- x_0, y- y_0, z- z_0>[tex] and the "set of all points such that the vector [tex]<x- x_0, y- y_0, z- z_0>[tex] is perpendicular to (the tangent plane to that vector containing ) must satisfy