Intersection of lines created by vectors

Question

If we have defined vectors $u$, $v$, $w$ and $r$, is there any elegant way to describe vector $s$ which lies on the intersection of lines created by pairs of vectors $(u, v)$ and $(w, r)$?

Diagram showing all vectors and lines

Answer 1

It has been a long time since I've approached such problems (so hopefully I don't make some trivial errors!) but I would think it would involve considering the following:

Based on your diagram you have two lines which move in the directions: $\tilde{w}-\tilde{r}$ and $\tilde{v}-\tilde{u}$, and they intersect at some vector $\tilde{s}$.

The vector representation of these lines are:

$L_1 = \tilde{v} + \lambda_1 \tilde{d}_1$

$L_2 = \tilde{w} + \lambda_2 \tilde{d}_2$

where $\lambda_1, \lambda_2$ are scalars, and $\tilde{d}_1$ and $\tilde{d}_2$ run parallel to the lines. These can be represented as $\tilde{d}_1 = \tilde{v}-\tilde{u}$ and $\tilde{d}_2 = \tilde{w}-\tilde{r}$, so that,

$L_1 = \tilde{v} + \lambda_1 (\tilde{v}-\tilde{u})$

$L_2 = \tilde{w} + \lambda_2 (\tilde{w}-\tilde{r})$

Thus we have two vector representations of our lines.

Thus since $\tilde{s}$ is where they intersect, you are looking for lambda values where $L_1 = L_2$. In otherwords $\tilde{s}$ is found by solving $\tilde{v} + \lambda_1 (\tilde{v}-\tilde{u}) - (\tilde{w} + \lambda_2 (\tilde{w}-\tilde{r})) = \tilde{0}$, for $\lambda_1$ and $\lambda_2$.

I'm not sure how much stronger of a representational form you are looking for.