Question

Find the shortest distance between the lines $\overrightarrow{r}=(4,-1,0)+k(1,2,3),k\in R$ & $\overrightarrow{r}=(1,-1,2)+k(2,4,-5),k\in R$.

Answer 1

Line (1) $\overrightarrow{r}=(4,-1,0)+(1,2,3)$.

Line (2) $\overrightarrow{r}=(1,-1,2)+k(2,4,-5)$

Shortest distance $=\left[\begin{array}{ccc}{x}_1-x_2& Y_1-Y_2& Z_1-Z_2\\ l_1& m_1& n_1\\ l_2& m_2& n_2\end{array}\right]$

$S>D=\left[\begin{array}{ccc}4-1& -1+1& 0-2\\ 1& 2& 3\\ 2& 4& -5\end{array}\right]=\left[\begin{array}{ccc}3& 0& -2\\ 1& 2& 3\\ 2& 4& -5\end{array}\right]$

$S.D=3\left[\begin{array}{cc}2& 3\\ 4& -5\end{array}\right]-0\left[\begin{array}{cc}1& 3\\ 2& -5\end{array}\right]-2\left[\begin{array}{cc}1& 2\\ 2& 4\end{array}\right]$

$S.D=3(-10-12)+0+0.$

$S.D=-66$

But distance is always positive.

then $S.D=66$