Cross product is proportional to sine of angle between vectors
==============================================================

The cross product of two vectors is a binary operation which produces a vector whose length is
proportional to the lengths of the given vectors and the sine of the angle between them.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Cross_product#Definition>`__.

.. py:currentmodule:: symplyphysics.laws.geometry.cross_product_is_proportional_to_sine_between_vectors

.. py:data:: cross_product_length

    Length of the cross product between :math:`\vec u` and :math:`\vec v`.

Symbol:
    :code:`norm(cross(u, v))`

Latex:
    :math:`\left \Vert \left[ \vec u, \vec v \right] \right \Vert`

Dimension:
    :code:`any_dimension`


.. py:data:: first_vector_length

    Length of :math:`\vec u`.

Symbol:
    :code:`u`

Latex:
    :math:`u`

Dimension:
    :code:`any_dimension`


.. py:data:: second_vector_length

    Length of :math:`\vec v`.

Symbol:
    :code:`v`

Latex:
    :math:`v`

Dimension:
    :code:`dimensionless`


.. py:data:: angle_between_vectors

    :attr:`~symplyphysics.symbols.basic.angle` between :math:`\vec u` and :math:`\vec v`.

Symbol:
    :code:`phi`

Latex:
    :math:`\varphi`

Dimension:
    :code:`angle`


.. py:data:: law

    :code:`norm(cross(u, v)) = u * v * sin(phi)`


    Latex:
        .. math::
            \left \Vert \left[ \vec u, \vec v \right] \right \Vert = u v \sin{\left(\varphi \right)}