Distance to a hyperplane

Topic: Support vector machines · more questions on this topic
A two-point machine ›

For w=(3,4)^\top and b=-5, compute the signed distance of (2,1) and (0,0) from the hyperplane w^\top x+b=0, and the margin between w^\top x+b=\pm 1.

Worked answer (click to show)

\lVert w\rVert=5. For (2,1), w^\top x+b=6+4-5=5, so the signed distance is 5/5=1. For (0,0), the signed distance is -5/5=-1: the origin is on the negative side, one unit away. The margin between w^\top x+b=\pm 1 is 2/\lVert w\rVert=2/5=0.4.