In another question here someone asks why a normal vector of a plane is simply a vector of the coefficients. For example:
For the plane $ 2x−y+3z=8 $, a normal vector is $(2,−1,3)$
The result is explained by saying if you take any 2 arbitrary points on the plane, you have $⟨x1−x2,y1−y2,z1−z2⟩⋅⟨2,−1,3⟩=0$ , and thus the coefficients must be the norm to the plane.
What I'm not understanding is why you can assume the coefficients dotted with the two points on the plane must equal zero. What is the intuition behind that assumption?
Best Answer
The fact is that every vectors in the plane is orthogonal to to the normal vector.
Since for all $(x_i,y_i,z_i)$:
$(x_1−x_2,y_1−y_2,z_1−z_2)$ represent a vector parallel to the plane
and since by the plane equation:
$⟨x_1−x_2,y_1−y_2,z_1−z_2⟩⋅⟨2,−1,3⟩=0$
thus
$n=(2,−1,3)$ is a normal vector to the plane.