# Proof of perpendicular distance of an observation from the Maximal Margin Hyperplane

I was reading about Maximal Margin Classifiers in "Introduction to Statistical Learning" and could not understand how is the perpendicular distance of an observation (which is a vector) from the hyperplane calculated ? I do know how it is done for a 2-D and 3-D space, but the formula (9.11) makes no sense to me. Shouldn't it use the MAGNITUDE ? Can anyone help me with this please ? The equation $$(9.11)$$ is after the dot product of the hyperplane and point $$x_i$$. The author has skipped vector normalization. In maximization problems, normalization does not matter because the magnitude of the normalization vector is a constant. The perpendicular distance of of a point $$x_i (x_{i1}, x_{i2}, ....,x_{1p})$$ from hyperplance is: $$\beta_0 + \beta_1x_{i1} + \beta_2x_{i2} + ... + \beta_px_{ip}$$
• observation point $$y_i$$ and calculated point (calculation is explained in image above) lie on same side of plane. This will ensure that our prediction is correct in terms of sign