Normalize a vector

Let's imagine we want  to normalize a vector  of three  variables;  that  is to compute a new  vector  of three values,  such that its size  is 1. Here is our ?rst attempt; it is a procedure that takes as input a list of three values, and gives a list of three numbers:

def normalize3(v):

return [v[0]/math.sqrt(v[0]**2+v[1]**2+v[2]**2), v[1]/math.sqrt(v[0]**2+v[1]**2+v[2]**2), v[2]/math.sqrt(v[0]**2+v[1]**2+v[2]**2)]

This is correct, but it looks pretty complicated. Let's start by giving that we're recalculating the denominator three times, and instead store the variable value.

def normalize3(v):

magv = math.sqrt(v[0]**2+v[1]**2+v[2]**2)

return [v[0]/magv,v[1]/magv,v[2]/magv]

Now,  we can see a repeated structure, of going through and dividing each component by magv.  Also, we seem that  the  computation of the  magnitude of a vector  is a useful  and  understandable operation in its own  right,  and  could probably be give  in its own method. That tends  to this procedure:

def mag(v):

return math.sqrt(sum([vi**2 for vi in v]))

def normalize3(v):

return [vi/mag(v) for vi in v]

This is especially nice, because  now,  in fact, it gives not just to vectors  of size  three.  So, it's both smaller and  more general than  what  we started with.  But, one of our real causes has snuck back in: we're recomputing mag(v) once for each element of the vector.  So, At last, here's a version that we're very happy with:9

def mag(v):

return math.sqrt(sum([vi**2 for vi in v]))

def normalize3(v):

magv = mag(v)

return [vi/magv for vi in v]