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

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

WBJEEWBJEE 2017Vector Algebra

Solution:

$\Delta=\begin{vmatrix}1 & -1 & 1 \\ 2 & -3 & 0 \\ 1 & 0 & 3\end{vmatrix}$
$=1(-9-0)-(-1) \cdot(6-0)+1(0-(-3))$
$=-9+6+3=0$
Since, $\Delta=0$
$\therefore $ Lines are coplanar.