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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 \(R^3\) 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 must have 3 dimensions.

  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 or b does not equal 3.

  See also

  `inner`

    Inner product

  `outer`

    Outer product.

  `linalg.cross`

    An Array API compatible variation of `np.cross`.

  `ix_`

    Construct index arrays.

  Notes

  Supports full broadcasting of the inputs.

  Examples

  Vector cross-product.

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

  One vector with dimension 2.

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

  Both vectors with dimension 2.

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