Distances to lines and planes

Using dot products, we can compute distances between geometric objects.

1. Let ${\bf n}$ be the normal vector for a plane through the point $Q$. The distance from another point $P$ to this plane is $\displaystyle{\frac{|\overrightarrow{Q\,P} \cdot {\bf n}|}{\| {\bf n}\|}}$.
2. The distance from the plane $Ax+By+Cz=D$ to the point $P(x_1,y_1,z_1)$ is $\displaystyle{\frac{|Ax_1 + By_1 + C z_1 -D|}{\sqrt{A^2+B^2+C^2}}}$.
3. The distance from the plane $Ax + By + Cz = D$ to the parallel plane $Ax + By + Cz = D'$ is $\displaystyle{\frac{|D-D'|}{\sqrt{A^2+B^2+C^2}}}$
4. The distance from a line with vector equation ${\bf r}(t) = {\bf b} + t {\bf v}$ to a point $P$ with position vector ${\bf u}$ is $\displaystyle{\frac {\| {\bf v} \times ({\bf u}-{\bf b})\|}{\| \bf v \|}}$.
5. The smaller angle between the plane $A_1 x + B_1 y + C_1 z = D_1$ and the plane $A_2 x + B_2 y + C_2 z = D_2$ is either the angle between the normal vectors ${\bf n_1} = \langle A_1, B_1, C_1 \rangle$ and ${\bf n_2} = \langle A_2, B_2, C_2 \rangle$, or is $\pi$ minus that angle, whichever is less. The cosine of the angle between the planes is $\displaystyle{\frac{|{\bf n_1}\cdot {\bf n_2}|}{\|{\bf n_1}\| \|{\bf n_2}\|}}$.