Let $A(a1,a2,a3),B(b1,b2,b3),C(c1,c2,c3),D(d1,d2,d3)$ be four different points. Let $E(e1,e2,e3)$ be the point on intersect lines $AB$ and $CD$. How to find $e1,e2,e3$?
I tried this way: vector $\overrightarrow{AB}$ must be multiplied by some constant $p$ to get $\overrightarrow{AE}$, so $p\overrightarrow{AB}=\overrightarrow{AE}$. Similar, $q\overrightarrow{CD}=\overrightarrow{CE}$ for some constant $q$. Now, I do not see any relation between $p$ and $q$. How to find coordinates of point $E$?
let $g_{AB}$ the straight lin throug $AB$ then we have $$\vec{x}=[a_1,a_2,a_3]+t[b_1-a_1,b_2-a_2,b_3-a_3]$$ and the other line is given by $$\vec{x}=[c_1,c_2,c_3]+s[d_1-c_1,d_2-c_2,d_3-c_3]$$ then solve the system $$\vec{x_t}=\vec{x_s}$$