If I understand your notation, the unit vector of maximum descent of a plane can be written as $$ (\cos\alpha\cos\theta, \cos\alpha\sin\theta, -\sin\alpha), $$ where $\theta$ is the azimuth and $\alpha$ is maximum dip. A horizontal unit vector perpendicular to that and lying along the plane can then be written as $$ (-\sin\theta,\cos\theta,0) $$ and the cross product of these two vectors gives the unit vector normal to the plane: $$ \hat n=(\sin\alpha\cos\theta, \sin\alpha\sin\theta, \cos\alpha). $$ The angle $\phi_{12}$ between two planes can be computed by the dot product of their normal vectors: $$ \cos\phi_{12}=\hat n_1\cdot\hat n_2= \sin\alpha_1\sin\alpha_2\cos(\theta_1-\theta_2)+\cos\alpha_1\cos\alpha_2. $$