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

Equation of $\overrightarrow{AB}=\frac{x-1}{3}=\frac{y+2}{2}=\frac{z+1}{-2}$

$\overrightarrow{CD}=\frac{x-1}{1}=\frac{y-2}{-6}=\frac{z+1}{-4}$

Distance between them is given by $\left|\begin{array}{ccc}{x}_1-x_2& y_1-y_2& z_1-z_2\\ l_1& m_1& n_1\\ l_2& m_2& n_2\end{array}\right|$

$=\left|\begin{array}{ccc}0& 4& 0\\ 3& 2& -2\\ 1& -6& -4\end{array}\right|$

$=(-4)(-12+2)=40$.