My Question is: How do I math the intersection($P5$) of $\overline{P3,P0}$ and $\overline{P7,P8}$ ?
I'm trying to find the intersection of two lines in a 3d space, (XYZ) on a plane.
I have a plane, it is formed by 3 points
$$ \begin{bmatrix} P1\\\hline 1\\ 1\\ 1\\ \end{bmatrix}, \begin{bmatrix} P0\\\hline -1\\ .345\\ 1\\ \end{bmatrix}, \begin{bmatrix} P3\\\hline 1\\ .275\\ -1\\ \end{bmatrix} $$
I also have 2 other points on that plane
$$ \begin{bmatrix} P7\\\hline .420\\ .687\\ .661\\ \end{bmatrix} \begin{bmatrix} P8\\\hline .295\\ .555\\ .410\\ \end{bmatrix} $$
Now if I draw a line though $\overline{P7,P8}$, it should be intersecting the line $\overline{P0,P3}$, at
$$ \begin{bmatrix} P5\\\hline .060\\ .308\\ -.060\\ \end{bmatrix} $$
(consequently, it should also intersect line $\overline{P1,P0}$ somewhere (.589,.866,1) )
My Question is: How do I math the intersection($P5$) of $\overline{P3,P0}$ and $\overline{P7,P8}$ ?
The vector for a general point on the line $P_7P_8$ is $$\lambda(P_7-P_8)+P_8=\lambda\begin{pmatrix}0.125\\0.132\\0.251\end{pmatrix}+\begin{pmatrix}0.295\\0.555\\0.410\end{pmatrix},$$ where $\lambda$ is any real number. You can think of this formula as starting at the point $P_8$, then shifting your position by a scalar multiple of the vector $\overrightarrow{P_8P_7}$.
Likewise the equation for a general point on line $P_0P_3$ is $$\mu(P_3-P_0)+P_0=\mu\begin{pmatrix}2\\-0.07\\-2\end{pmatrix}+\begin{pmatrix}1\\0.275\\-1\end{pmatrix}.$$ Equating these two expressions should give you three equations (one for each component) for $\lambda$ and $\mu$, which you can then solve.