Q.Find the equation of the plane which passes through the point $(3,4,-1)$ and is parallel to the plane $2x-3y+5z+7=0$. Also, find the distance between the two planes.

Four closely related "how far / how tilted" computations, all built from a plane's normal $\vec n$ or a line's direction $\vec b$.

Angle between two planes = angle between their normals: $\theta=\cos^{-1}\!\left(\dfrac{|\vec n_1\cdot\vec n_2|}{|\vec n_1||\vec n_2|}\right)$; perpendicular iff $\vec n_1\cdot\vec n_2=0$, parallel iff $\vec n_1=\lambda\vec n_2$. …