For an indexed angular variable [math]\theta_i[/math], known in quantum field theories as the phase, the direction cosines of a given vector [math]\mathbf{r}_i[/math] is given by [math]l_i=\cos \theta_i[/math] and [math]m_i=\sin \theta_i[/math]. Therefore, every non-zero indexed vector [math]\mathbf{r}_i[/math] is associated with a pair of these direction cosines. Although the polar variables [math]\theta_i[/math] and [math]\mathbf{r}_i[/math] are independent of each other, as shown by separability theorem http://en.wikipedia.org/wiki/Separation_of_variables, when the absolute magnitude of [math]\mathbf{r}_i[/math] vanishes (for zero or null vectors), the pair of direction cosines also vanishes. However, parallelism and perpendicularity theorem indicate that these can be used to demonstrate the existence of parallel displacement and orthogonal vectors including their spatio-temporal orientations. For the same orientation the sum of the corresponding product of direction cosines is unity, [math]l_1l_2+m_1m_2=1[/math]. For opposite orientation, [math]l_1l_2+m_1m_2=-1[/math]. For orthogonal vectors, [math]l_1l_2+m_1m_2=0[/math]. Furthermore, the triangular area bounded by three vectors is given by
[math]A=\frac{1}{2}\left|\begin{array}{ccc}x_1&y_1&1\\x_ 2&y_2&1\\x_3&y_3&1\end{array}\right|[/math]
Where [math]x_i=r_i\cos \theta_i[/math] and [math]y_i=\sin \theta_i[/math]. This area vanishes if [math]r_1=0[/math] or [math]r_2=0[/math] or [math]r_3=0[/math]. Moreover, a necessary condition for area to exist is [math]\theta_1+\theta_2+\theta_3=180^{\circ}[/math].


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