Orthogonalizing three vectors

In [1]: 
import numpy as np
import numpy.linalg as la
In [3]: 
np.random.seed(13)
xorig = np.random.randn(200)
yorig = np.random.randn(200)
zorig = np.random.randn(200)

Orthonormalize $x$ and $y$ as we know:

In [4]: 
x = xorig/la.norm(xorig)
In [6]: 
y = yorig
y = y - x.dot(y)*x
y = y / la.norm(y)

Check:

In [7]: 
print(la.norm(x))
print(la.norm(y))
print(x.dot(y))

1.0
1.0
-4.07660016855e-17

Now what to with $z$?

In [8]: 
z = zorig
z = z - np.dot(z, x)*x - np.dot(z,y)*y
z = z / la.norm(z)

Check:

In [9]: 
print(la.norm(z))
print(x.dot(z))
print(y.dot(z))

1.0
-1.38777878078e-17
-3.68628738645e-17