Equation of the Common Perpendicular

Equation of the Common Perpendicular

For two skew lines:

$$\vec r=\vec a+\lambda\vec b$$

$$\vec r=\vec c+\mu\vec d$$

the common perpendicular has direction:

$$\vec b\times\vec d$$

and can be represented as:

$$\vec r=\vec p+t(\vec b\times\vec d)$$

where $$\vec p$$ is the point of contact on either line.

Usage

- Used when the actual common perpendicular is required rather than only its length.