Let the right triangle hypothenuse be aligned with the coordinate system x-axis. The vector loop closure equation running counter-clockwise 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
$$-c{\bold e}x = a{\bold e}\alpha + b\tilde{\bold e}_\alpha$$
and squaring
finally results in the Pythagorean theorem (2)
(2)
Introducing the hypothenuse segments
segment p | segment q | height h | area |
---|---|---|---|