# numpy.cross#

numpy.cross(a, b, axisa=- 1, axisb=- 1, axisc=- 1, axis=None)[source]#

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
aarray_like

Components of the first vector(s).

barray_like

Components of the second vector(s).

axisaint, optional

Axis of a that defines the vector(s). By default, the last axis.

axisbint, optional

Axis of b that defines the vector(s). By default, the last axis.

axiscint, 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.

axisint, 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
cndarray

Vector cross product(s).

Raises
ValueError

When the dimension of the vector(s) in a and/or b does not equal 2 or 3.

inner

Inner product

outer

Outer product.

ix_

Construct index arrays.

Notes

New in version 1.9.0.

Supports full broadcasting of the inputs.

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)
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]])