Explanation of this code: Determining if a point is on which side of a line

Question

Source: http://datasciencelab.wordpress.com/2014/01/10/machine-learning-classics-the-perceptron/

"The general equation of a line given two points in it, (x1,y2) and (x2,y2), is A + Bx + Cy = 0 where A, B, C can be written in terms of the two points. Defining a vector V = (A, B, C), any point (x,y) belongs to the line if V'x = 0, where x = (1,x,y). Points for which the dot product is positive fall on one side of the line, negatives fall on the other."

I don't quite understand how it works. Also, this line in particular:

self.V = np.array([xB*yA-xA*yB, yB-yA, xA-xB])

Why is Bx determined by yb-ya?

For what its worth, I'm learning Linear Algebra, so I'm quite familiar with the mathematical concept (I realize it is meant to be a normal), but how it is done escapes me.

Answer 1

Why is B determined by yb - ya? (or rather y2 - y1 for our example)

The text assumes the line equation to be: A + B*x + C*y = 0

Let's say we have two points from that line, P1(x1, y1) and P2(x2, y2)

Using the two-points form of the line equation, you will get y - y1 = [(y2 - y1)/(x2 - x1)] * (x - x1) (based on P1and P2)

The same equation, can be developed into [(x2 -x1)*y1 + (y1 - y2)*x1] + (y2 - y1) * x + (x1 - x2) * y = 0

Looking at A + B*x + C*y = 0, you see that:

• A, is [(x2 -x1)*y1 + (y1 - y2)*x1] = x2*y1 - y2*x1

• B, the coefficient of x is (y2 - y1)

• C, the coefficient of y is (x1 - x2)

hence the value np.array([A, B, C]) in the source code appears as np.array([xB*yA - xA*yB, yB - yA, xA - xB])