How to find intersection point of two 3D lines

Question

I know the $x,y,z$ values of the 4 points that create two lines. I also know the $x,y$ value of the crossing point. How do I get the $z$ of the crossing point?

Answer 1

given four points $\mathbf{a,b,c,d}$, form the parametric equations

$\mathbf{a}+t \mathbf{(b-a)}$ and $\mathbf{c}+s \mathbf{(d-c)}$, where $s,t \in \mathbb R$ are parameters.

This will look like $\ell_1:=\{(a_1,a_2,a_3)+t(b_1-a_1,b_2-a_2,b_3-a_3) \mid t \in \mathbb R\}$, and analogous for $\ell_2$.

If you know the $x,y$ values for the intersection, then there is a system of linear equations:

$x=a_1+t(b_1-a_1)=c_1+s(d_1-c_1)$ and $y=a_2+t(b_2-a_2)=c_1+s(d_2-c_2)$. Solve these for $s,t$ and plug them back into the parametric equations to get the $z$ value.