Question

Prove that lines $\frac{x-1}{3}=\frac{y+1}{2}=\frac{z-1}{5}$ and $\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{-2}$ are skew lines.

Answer 1

From line $L:\frac{x-1}{3}=\frac{y-(-1)}{2}=\frac{z-1}{5}$
We get Point $A\left(\overline{b}\right)=(1,-1,1)$ and direction is $\overline{l}=(3,2,5)$ and from line
$M:\frac{x+2}{4}=\frac{y-1}{3}=\frac{z+1}{-2}$
We get point $B\left(\overline{b}\right)=(-2,1,-1)$ and direction is $\overline{m}=(4,3,-2)$
Now, $\overline{l}\times \overline{m}=(3,2,5)\times (4,3,-2)=(\left|\begin{array}{cc}2& 5\\ 3& -2\end{array}\right|,-\left|\begin{array}{cc}3& 5\\ 4& -2\end{array}\right|,\left|\begin{array}{cc}3& 2\\ 4& 3\end{array}\right|)$
$=(3,-2,2).3(-19,26,1)=57-52+2$
$\therefore \overline{l}\times \overline{m}=107\ne 0$
Hence the given lines are skew lines