generating two orthogonal vectors that are orthogonal to a particular direction

Question

What is the simplest and most efficient ways in numpy to generate two orthonormal vectors a and b such that the cross product of the two vectors equals another unit vector k, which is already known?

I know there are infinitely many such pairs, and it doesn't matter to me which pairs I get as long as the conditions axb=k and a.b=0 are satisfied.

Answer 1

This will do:

>>> k  # an arbitrary unit vector k is not array. k is must be numpy class. np.array
np.array([ 0.59500984,  0.09655469, -0.79789754])

To obtain the 1st one:

>>> x = np.random.randn(3)  # take a random vector
>>> x -= x.dot(k) * k       # make it orthogonal to k
>>> x /= np.linalg.norm(x)  # normalize it

To obtain the 2nd one:

>>> y = np.cross(k, x)      # cross product with k

and to verify:

>>> np.linalg.norm(x), np.linalg.norm(y)
(1.0, 1.0)
>>> np.cross(x, y)          # same as k
array([ 0.59500984,  0.09655469, -0.79789754])
>>> np.dot(x, y)            # and they are orthogonal
-1.3877787807814457e-17
>>> np.dot(x, k)
-1.1102230246251565e-16
>>> np.dot(y, k)
0.0

Answer 2

Sorry, I can't put it as a comment because of a lack of reputation.

Regarding @behzad.nouri's answer, note that if k is not a unit vector the code will not give an orthogonal vector anymore!

The correct and general way to do so is to subtract the longitudinal part of the random vector. The general formula for this is here

So you simply have to replace this in the original code:

>>> x -= x.dot(k) * k / np.linalg.norm(k)**2