I frequently have to re-implement projecting vectors onto other vectors, and rotating vectors about other vectors. So here we go. Public domain.

rotateVectors.py

import unittest
import numpy as np
import numpy.linalg as linalg


def makeUnit(x):
    """Normalize entire input to norm 1. Not what you want for 2D arrays!"""
    return x / linalg.norm(x)


def xParV(x, v):
    """Project x onto v. Result will be parallel to v."""
    # (x' * v / norm(v)) * v / norm(v)
    # = (x' * v) * v / norm(v)^2
    # = (x' * v) * v / (v' * v)
    return np.dot(x, v) / np.dot(v, v) * v


def xPerpV(x, v):
    """Component of x orthogonal to v. Result is perpendicular to v."""
    return x - xParV(x, v)


def xProjectV(x, v):
    """Project x onto v, returning parallel and perpendicular components
    >> d = xProject(x, v)
    >> np.allclose(d['par'] + d['perp'], x)
    True
    """
    par = xParV(x, v)
    perp = x - par
    return {'par': par, 'perp': perp}


def rotateAbout(a, b, theta):
    """Rotate vector a about vector b by theta radians."""
    # Thanks user MNKY at http://math.stackexchange.com/a/1432182/81266
    proj = xProjectV(a, b)
    w = np.cross(b, proj['perp'])
    return (proj['par'] +
            proj['perp'] * np.cos(theta) +
            linalg.norm(proj['perp']) * makeUnit(w) * np.sin(theta))


class TestRotation(unittest.TestCase):
    def test_parallel2D(self):
        x = np.array([0, 1.0])
        v = np.array([-1, 1.0])
        vhat = v / linalg.norm(v)
        expected = vhat * np.cos(np.pi / 4)
        self.assertTrue(np.allclose(expected, xParV(x, v)))

    def test_perp3D(self):
        x = np.array([1, 1, 1.0])
        v = np.array([1, 0, 0.0])
        expected = np.array([0, 1, 1.0])
        self.assertTrue(np.allclose(expected, xPerpV(x, v)))

    def test_rotate3D(self):
        toRotate = np.array([1, 1, 1.0])
        about = np.array([0, 1, 0.0])
        actual = rotateAbout(toRotate, about, np.pi / 180 * 90)
        expected = np.array([1, 1, -1.0])
        self.assertTrue(np.allclose(expected, actual))


if __name__ == '__main__':
    unittest.main()