Problems from Distance between two straight lines in space

Development:

We start by determining the relative position of the straight lines.

First we verify that the governing vectors are not linearly dependent: $$\begin{array}{l}\overrightarrow{v}=(2,-1,1)\\ {\overrightarrow{v}}^{\prime }=(1,3,-2)\end{array}\}\Rightarrow {\displaystyle \frac{2}{1}}\ne {\displaystyle \frac{-1}{3}}\ne {\displaystyle \frac{1}{-2}}$$

The straight lines $r$ and $r^{\prime }$ intersect or cross.

We take a point $A$ of $r$ and a point $A^{\prime }$ of $r^{\prime }$, and see if $\{\overrightarrow{A{A}^{\prime }},\overrightarrow{v},{\overrightarrow{v}}^{\prime }\}$ are linearly dependent or independent: $$\begin{array}{l}A=(2,1,3)\\ A^{\prime }=(-1,-1,4)\end{array}\}\Rightarrow \overrightarrow{A{A}^{\prime }}=(-3,-2,1)$$

$$\left|\begin{array}{ccc}2& 1& -3\\ -1& 3& -2\\ 1& -2& 1\end{array}\right|=0\Rightarrow \text{rank}(\{\overrightarrow{A{A}^{\prime }},\overrightarrow{v},{\overrightarrow{v}}^{\prime }\})=0$$

Therefore the straight lines r and r' intersect and $\text{d}(r,r^{\prime })=0$.

Solution:

$\text{d}(r,r^{\prime })=0$

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