Relativistic velocity tangential to movement
============================================

Consider two inertial reference frames: one fixed (lab frame) and one tied to the moving object
(proper frame). The proper frame is moving with some velocity :math:`\vec v` relative to the lab
frame. According to the theory of special relativity, the velocity of the object relative to lab
frame is *not* equal to the sum of its velocity in the proper frame and the velocity of the proper
frame relative to the lab frame.

**Notes:**

#. One can get the same expression for :math:`{\vec u'}_\perp` in terms of :math:`\vec u` by
   replacing :math:`\vec v` with :math:`-{\vec v}`. This is essentially the inverse Lorentz
   transformation from lab frame to proper frame that uses the fact that the lab frame can be
   viewed as moving with velocity vector :math:`-{\vec v}` relative to the proper frame.

**Conditions:**

#. Works in special relativity.

**Links:**

#. `Wikipedia <https://en.wikipedia.org/wiki/Velocity-addition_formula#General_configuration>`__.

..
    TODO: rename file

.. py:currentmodule:: symplyphysics.laws.relativistic.vector.relativistic_velocity_parallel_to_movement

.. py:data:: tangential_velocity_in_lab_frame

    Component of the velocity vector relative to the lab frame tangential to :math:`\vec v`. See
    :attr:`~symplyphysics.symbols.classical_mechanics.speed`.

Symbol:
    :code:`u_t`

Latex:
    :math:`{\vec u}_\text{t}`

Dimension:
    :code:`velocity`


.. py:data:: velocity_in_proper_frame

    Velocity vector relative to the proper frame. See :attr:`~symplyphysics.symbols.classical_mechanics.speed`.

Symbol:
    :code:`u'`

Latex:
    :math:`{\vec u'}`

Dimension:
    :code:`velocity`


.. py:data:: proper_frame_velocity

    Velocity vector of the proper frame relative to the lab frame. See :attr:`~symplyphysics.symbols.classical_mechanics.speed`.

Symbol:
    :code:`v`

Latex:
    :math:`{\vec v}`

Dimension:
    :code:`velocity`


.. py:data:: law

    :code:`u_t = v * (dot(u', v) * dot(v, v)^(-1) + 1) * (1 + dot(u', v) / c^2)^(-1)`


    Latex:
        .. math::
            {\vec u}_\text{t} = {\vec v} \left(\left( {\vec u'}, {\vec v} \right) \left( {\vec v}, {\vec v} \right)^{-1} + 1\right) \left(1 + \frac{\left( {\vec u'}, {\vec v} \right)}{c^{2}}\right)^{-1}