Support vector machines - separating hyperplane question

Answer 1

Imagine a plane in a 3d coordinate system. To describe it, you need a normal vector N of that plane and the distance D of the plane to the origin. For simplicity, assume the normal vector has unit length. Then the equation for that plane is x.N - D = 0.

Explanation: x.N can be visualized as a projection of x on the normal vector N. The result is the length of vector x parallel to N. If this length equals D, the point x is on the plane.