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I stumbled on the following problem:

Given $u^TAv = m$, $A^T A = I$ for $A = [x, -y; y, x]$, $u, v \in \mathbb{R}^2$ and $m \in \mathbb{R}$ , find $A$.

The problem emerges from the modelling from closed chain mechanism on theoretical mechanics. I appreciate your interest on an $\textbf{elegant}$ solution.

Best regards, Bruno

Bernard
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Bruno Lobo
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1 Answers1

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$A$ is a rotation matrix. Rewriting in complex numbers,

$$\Re(u^*e^{i\theta}v)=|u^*v|\cos(\angle(u^*v)+\theta)=m$$

gives you $\theta=\arccos\dfrac{m}{|u^*v|}-\angle(u^*v)$.