Question

If $A(1,2,-1),B(4,0,-3),C(1,2,-1),D(2,-4,-5)$, find the distance between $AB$ and $CD$.

Answer 1

Direction ratio of $AB=3,-2,-2$

Direction ratio of $\overline{CD}=1,-6,-4$

Distance between lines having D.R's $a,b,c$ and $d,e,f$ and passing through $(x_1,y_1,z_1),(x_2,y_2,z_2)$ is given by, $d=\left|\begin{array}{ccc}{x}_1-x_2& y_1-y_2& x_1-x_2\\ a& b& c\\ d& e& f\end{array}\right|$

let $A=x_1,y_1,z_1\to d=\left|\begin{array}{ccc}0& 0& 0\end{array}\right|$

& $C=x_2,y_2,z_2$

as one row is zero,

$d=0$