Q.Find the angle between the two planes $3x-6y+2z=10$ and $2x+2y-2z=15$.

Two planes are inclined to each other exactly as much as their normal vectors are inclined to each other, so the angle between two planes is defined through the angle between their normals rather than through anything measured inside the planes themselves. Planes $\vec r\cdot\vec n_1=d_1$ and $\vec r\cdot\vec n_2=d_2$ are perpendicular precisely when $\vec n_1\cdot\vec n_2=0$; in Cartesian form, for $a_1x+b_1y+c_1z+d_1=0$ and $a_2x+b_2y+c_2z+d_2=0$, this becomes $a_1a_2+b_1b_2+c_1c_2=0$. When the planes are not perpendicular, the angle between them is, by convention, the acute angle between the normals, since two intersecting planes actually meet at a pair of supplementary angles and only the acute one is reported: $\cos\theta …