In the $2$nd page [here](http://www.damtp.cam.ac.uk/user/tong/dynamics/three.pdf) of David Tong's notes on the Motion of Rigid Bodies, at the bottom there is a claim about the unique existence of a certain time-dependent orthonormal matrix $R(t)$ whose $9$ scalar components $R\_{ab}(t)$ (where $a,b\in\{1,2,3\}$) are postulated to be capable of relating the *fixed space frame* basis vectors $\tilde{\textbf e}\_a$ (where $a=1,2,3$) to the *moving body frame* basis vectors $\textbf e\_a(t)$ (where $a=1,2,3$) via a simple scalar proportionality relation:
$$\textbf e\_a(t)=R\_{ab}(t)\tilde{\textbf e}\_b\space\space \text{ for all}\space a,b\in\{1,2,3\}$$
But this does not seem possible because it is suggesting that each of the moving body frame basis vectors $\textbf e\_a(t)$ must be aligned/collinear with each of the fixed space frame basis vectors $\tilde{\textbf e}\_a$, hence all of the vectors are collinear! What is the issue in my understanding here?