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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 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). Ignored if both input vectors have dimension 2, as the return is scalar. 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.

  `linalg.cross`

    An Array API compatible variation of `np.cross`, which accepts (arrays of) 3-element vectors only.

  `ix_`

    Construct index arrays.

  Notes

  Supports full broadcasting of the inputs.

  Dimension-2 input arrays were deprecated in 2.0.0. If you do need this functionality, you can use:

      def cross2d(x, y):
          return x[..., 0] * y[..., 1] - x[..., 1] * y[..., 0]

  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]
      >>> 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)
      array(-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]])