Question

Three lines are drawn from the origin $O$ with direction cosines proportional to $(1,-1,1),(2,-3,0)$ and $(1,0,3),$ Then three lines are

• A. not coplanar
• B. coplanar
• C. perpendicular to each other
• D. coincident

Answer 1

The correct option is B coplanar

Let $\Delta =\left|\begin{array}{ccc}1& -1& 1\\ 2& -3& 0\\ 1& 0& 3\end{array}\right|$
If line are perpendicular to each other then $\Delta \ne 0$
or If line are coplaner then $\Delta =0$
$\Delta =\left|\begin{array}{ccc}1& -1& 1\\ 2& -3& 0\\ 1& 0& 3\end{array}\right|$

$C_2\to C_2+C_1{C}_3\to C_3-C_1$
$\left|\begin{array}{ccc}1& 0& 0\\ 2& -1& -2\\ 1& 1& 2\end{array}\right|=1(-2+2)=0$
Hence three lines are coplanar