Angle Between Two Planes

The angle between two planes is defined to be the angle between their normals.

Theorem 6.18 (vector form). The acute angle $\theta$ between planes $\vec r\cdot\vec n_1=p_1$ and $\vec r\cdot\vec n_2=p_2$ is

Remark.

1. The planes are perpendicular iff $\vec n_1\cdot\vec n_2=0$.
2. The planes are parallel iff $\vec n_1=\lambda\vec n_2$ for some scalar $\lambda$.
3. A plane parallel to $\vec r\cdot\vec n=p$ has equation $\vec r\cdot\vec n=k$ for any constant $k\in\mathbb R$ (same normal, different "offset"). Theorem 6.19 (Cartesian form). For planes $a_1x+b_1y+c_1z+d_1=0$ and $a_2x+b_2y+c_2z+d_2=0$ (normals $(a_1,b_1,c_1),(a_2,b_2,c_2)$):

   $\theta=\cos^{-1}\left(\frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\,\sqrt{a_2^2+b_2^2+c_2^2}}\right).$

   Remark. …