Strike, dip, and rake directions in focal mechanism

Vincent S. Cronin wrote A Primer on Focal Mechanism Solutions for Geologists (2010). My interest in his work is in his descriptions of fault vectors and angles. I need to use these standard names if I wish to write a paper on Focal Mechanism of Earthquakes.

Based on Cronin's diagram in p. 6, the reference strike, which we shall denote by $\mathbf e_{strike}$ appears to be the direction of the fault line as seen from a drone camera flying overhead high above the fault. The dip vector, which we shall denote by $\mathbf e_{dip}$, is a vector perpendicular to the reference strike $\mathbf e_{strike}$ and lies along the fault plane. The direction of $\mathbf e_{dip}$ is downward along the fault in Cronin. In this way, if we define the rake $\mathbf e_{rake}$ as the direction of the movement of the one side of the fault, then $\mathbf e_{rake}$ can be expressed as a rotation of $\mathbf e_{strike}$ about $\mathbf e_{dip}\times\mathbf e_{strike}$ by an angle $\theta_{rake}$, which is positive if the rotation is counterclockwise and negative if clockwise, following the right hand rule. These considerations results to the following expression for the rake direction $\mathbf e_{rake}$:
\begin{equation}
\mathbf e_{rake} = \mathbf e_{strike}\cos\theta_{rake} - \mathbf e_{dip}\sin\theta_{dip}.
\end{equation}
But even then, there is an uncertainty in the definition of the strike direction $\mathbf e_{strike},$ since $-\mathbf e_{strike}$ can also be the initial definition. Perhaps, the strike direction should be defined as that which makes an acute angle with respect to the north direction.