Orthogonalizing vectors

In [4]: 
import numpy as np
import matplotlib.pyplot as pt
In [25]: 
np.random.seed(13)
x = np.random.randn(2)
y = np.random.randn(2)
In [22]: 
pt.arrow(0, 0, x[0], x[1], color="blue")
pt.arrow(0, 0, y[0], y[1], color="red")
pt.xlim([-1, 1])
pt.ylim([-1, 1])
pt.gca().set_aspect("equal")
pt.grid()

Are those orgonal? How would we find out?

In [26]: 
x.dot(y)
Out[26]:
0.37226452785072001

Now use the formula to make ynew which is orthogonal to x:

In [27]: 
ynew = y - y.dot(x)/x.dot(x)*x
In [28]: 
pt.arrow(0, 0, x[0], x[1], color="blue")
pt.arrow(0, 0, ynew[0], ynew[1], color="red")
pt.xlim([-1, 1])
pt.ylim([-1, 1])
pt.gca().set_aspect("equal")
pt.grid()
In []: