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# numpy.cross

`numpy.``cross`(a, b, axisa=-1, axisb=-1, axisc=-1, axis=None)

  Return the cross product of two (arrays of) vectors.

  The cross product of a and b in is a vector perpendicular to both a and b. 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. Where the dimension of either a or b is 2, the third component of the input vector is assumed to be zero and the cross product calculated accordingly. In cases where both input vectors have dimension 2, the z-component of the cross product is returned.

  Parameters : | a : array_like Components of the first vector(s). b : array_like Components of the second vector(s). axisa : int, optional Axis of a that defines the vector(s). By default, the last axis. axisb : int, optional Axis of b that defines the vector(s). By default, the last axis. axisc : int, optional Axis of c containing the cross product vector(s). By default, the last axis. axis : int, optional If defined, the axis of a, b and c that defines the vector(s) and cross product(s). Overrides axisa, axisb and axisc.
  Returns : | c : ndarray Vector cross product(s).
  Raises : | ValueError When the dimension of the vector(s) in a and/or b does not equal 2 or 3.

  See also

  `inner`
    Inner product
  `outer`
    Outer product.
  `ix_`
    Construct index arrays.

  Examples

  Vector cross-product.

      >>> x = [1, 2, 3]
      >>> y = [4, 5, 6]
      >>> np.cross(x, y)
      array([-3,  6, -3])

  One vector with dimension 2.

      >>> x = [1, 2]
      >>> y = [4, 5, 6]
      >>> np.cross(x, y)
      array([12, -6, -3])

  Equivalently:

      >>> x = [1, 2, 0]
      >>> y = [4, 5, 6]
      >>> np.cross(x, y)
      array([12, -6, -3])

  Both vectors with dimension 2.

      >>> x = [1,2]
      >>> y = [4,5]
      >>> np.cross(x, y)
      -3

  Multiple vector cross-products. Note that the direction of the cross product vector is defined by the right-hand rule.

      >>> x = np.array([[1,2,3], [4,5,6]])
      >>> y = np.array([[4,5,6], [1,2,3]])
      >>> np.cross(x, y)
      array([[-3,  6, -3],
             [ 3, -6,  3]])

  The orientation of c can be changed using the axisc keyword.

      >>> np.cross(x, y, axisc=0)
      array([[-3,  3],
             [ 6, -6],
             [-3,  3]])

  Change the vector definition of x and y using axisa and axisb.

      >>> x = np.array([[1,2,3], [4,5,6], [7, 8, 9]])
      >>> y = np.array([[7, 8, 9], [4,5,6], [1,2,3]])
      >>> np.cross(x, y)
      array([[ -6,  12,  -6],
             [  0,   0,   0],
             [  6, -12,   6]])
      >>> np.cross(x, y, axisa=0, axisb=0)
      array([[-24,  48, -24],
             [-30,  60, -30],
             [-36,  72, -36]])