It is given that inner product $$ \left\langle a(t),b(t)\right\rangle =0,\quad \forall t\in[0,T] $$ where $a(t), b(t)\in \mathbb{R}^n$.
If $\dot{a}(t)$ is known, is there a way to find an expression for $\dot{b}(t)$ ? If it helps $\dot{a}(t)=-Aa(t)$, $A\in \mathbb{R}^{n\times n}$
So far I got only this $$ \left\langle a(t),\dot{b}(t)\right\rangle +\left\langle \dot{a}(t),b(t)\right\rangle =0 $$
$\left\langle a(t),v\right\rangle = 0$ define hyperplanes through the origin. Depending on the matrix $A$ it may rotate or not, take for example $A=I$.
Alternatively, think of $a(t)$ as tangents along a curve $\gamma(t)$, $\dot\gamma(t)=a(t)$, then $b(t)$ is the moving normal to the curve. In any case finding $b(t)$ would indeed require some more information.