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# Triangle Let the right triangle hypothenuse be aligned with the coordinate system *x-axis*. The vector loop closure equation then reads $$a{\bold e}_\alpha + b\tilde{\bold e}_\alpha + c{\bold e}_x = \bold 0$$ (1) with $${\bold e}_\alpha = \begin{pmatrix}\cos\alpha\\ \sin\alpha\end{pmatrix} \quad and \quad {\tilde\bold e}_\alpha = \begin{pmatrix}-\sin\alpha\\ \cos\alpha\end{pmatrix}$$ Resolving for the hypothenuse part $c{\bold e}_x$ in the loop closure equation (1) $$-c{\bold e}_x = a{\bold e}_\alpha + b\tilde{\bold e}_\alpha$$ and squaring > finally results in the Pythagorean theorem (2) > > $$ c^2 = a^2 + b^2 $$ (2)