Question

Consider the following vectors (essentially2x1 matrices):

a = sc.array([[1], [2], [3]])
>>> a
[[1]
 [2]
 [3]]

b = sc.array([[4], [5], [6]])
>>> b
[[4]
 [5]
 [6]]

The cross product of these vectors can be calculated using numpy.cross(). Why does this not work:

import numpy as np 

np.cross(a, b)
ValueError: incompatible dimensions for cross product
(dimension must be 2 or 3)

but this does?:

np.cross(a.T, b.T)
[[-3  6 -3]]

Answer 1

To compute the cross product using numpy.cross, the dimension (length) of the array dimension which defines the two vectors must either by two or three. To quote the documentation:

If a and b are arrays of vectors, the vectors
are defined by the last axis of a and b by default, and these axes
can have dimensions 2 or 3.

Note that the last axis is the default. In your example:

In [17]: a = np.array([[1], [2], [3]])

In [18]: b = np.array([[4], [5], [6]])

In [19]: print a.shape,b.shape
(3, 1) (3, 1)

the last axis is only of length 1, so the cross product is not defined. However, if you use the transpose, the length along the last axis is 3, so it is valid. You could also do:

In [20]: np.cross(a,b,axis=0)
Out[20]: 
array([[-3],
       [ 6],
       [-3]])

which tells cross that the vectors are defined along the first axis, rather than the last axis.