Normal vector between two parallel lines

Question

Is there a way to calculate the normal vector of two parallel lines, without calculating the length or the points?

Answer 1

Suppose the lines are those parallel to $\bf v$ and through the points ${\bf x}, {\bf y}$, so that the parameterizations are $t \mapsto {\bf x} + t {\bf v}$ and $t \mapsto {\bf y} + t {\bf v}$. Then, to recover the normal vector between the lines, we can simply project the difference ${\bf y} - {\bf x}$ onto the plane orthogonal to ${\bf v}$, giving the vector

$$({\bf y} - {\bf x}) - \operatorname{proj}_{\bf v}({\bf y} - {\bf x}) = ({\bf y} - {\bf x}) - \left(\frac{({\bf y} - {\bf x}) \cdot {\bf v}}{{\bf v} \cdot {\bf v}}\right) {\bf v} .$$

Of course, this formula simplifies nicely if we first normalize $\bf v$ to have unit length.

In $\Bbb R^3$, we can alternatively write this using the cross product. A quick geometric sketch shows that if we normalize $\bf v$ to produce a unit vector $\hat{\bf v}$, then the normal vector between the lines is $$ \hat{\bf v} \times [({\bf y} - {\bf x}) \times \hat{\bf v}] . $$