Angle Between Two Planes

Acute dihedral angle between planes a₁x+b₁y+c₁z+d₁=0 and a₂x+b₂y+c₂z+d₂=0. The angle between planes equals the angle between their normals (or its supplement — take the acute one).

The dihedral angle between two planes is the angle between their normals (or its supplement — we take the acute one).

For planes $\Pi_1: a_1x+b_1y+c_1z+d_1=0$ and $\Pi_2: a_2x+b_2y+c_2z+d_2=0$:

Normals: $\mathbf{n_1} = (a_1,b_1,c_1)$ and $\mathbf{n_2} = (a_2,b_2,c_2)$.

$$\cos\theta = \frac{|\mathbf{n_1}\cdot\mathbf{n_2}|}{|\mathbf{n_1}||\mathbf{n_2}|} = \frac{|a_1a_2+b_1b_2+c_1c_2|}{\sqrt{a_1^2+b_1^2+c_1^2}\cdot\sqrt{a_2^2+b_2^2+c_2^2}}$$

Perpendicular planes: $a_1a_2+b_1b_2+c_1c_2 = 0$.

Parallel planes: $a_1/a_2 = b_1/b_2 = c_1/c_2$ (normals are parallel).

In vector form: For planes $\mathbf{r}\cdot\mathbf{n_1}=d_1$ and $\mathbf{r}\cdot\mathbf{n_2}=d_2$:

$$\cos\theta = \frac{|\mathbf{n_1}\cdot\mathbf{n_2}|}{|\mathbf{n_1}||\mathbf{n_2}|}$$