Distance of a point from a line passing through two points.

Question

Say I have these 2 lines represented as parametric equations like so:

$L1 : x= 1 + t, y = 0, z = 2 - t$

$L2 : x = t , y = t, z = 1 + t$

My question is:

Is there a way to find the distance (minimal) between those 2 lines only by using the formula listed below?

$$\frac{|| AB \times AP || }{ ||AB||}$$ (Distance of a point from a line passing through two points.)

$A$ is a point on $L1$

$B$ is a point on $L1$

$P$ is a point on $L2$

Answer 1

No, there is no way. $$ \frac{\lVert AB \times AP \rVert}{\lVert AB \rVert} = \frac{\lVert AB \rVert \lVert AP \rVert \sin \alpha}{\lVert AB \rVert} = \lVert AP \rVert \sin \alpha $$ This formula only depends on the distance between $A$ and $P$ and that angle between the first line and $AP$ which is the shortest connection between $A$ and $P$, so it leaves out the direction of the second line.

We could rotate the second line around $P$ without changing the value of that formula, while that rotation might change the minimal distance between both lines.

For an example minimizing the (squared) distance over two parameters see here.