The direction cosines of three lines passing | vedclass

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(A) Three lines passing through the origin with direction cosines $(l_1, m_1, n_1)$,$(l_2, m_2, n_2)$,and $(l_3, m_3, n_3)$ are coplanar if they lie i

Vedclass · Accepted Answer

$\left| \begin{array}{ccc} l_1 & n_1 & m_1 \ l_2 & n_2 & m_2 \ l_3 & n_3 & m_3 \end{array} ight| = 0$

Vedclass · Answer

(A) Three lines passing through the origin with direction cosines $(l_1, m_1, n_1)$,$(l_2, m_2, n_2)$,and $(l_3, m_3, n_3)$ are coplanar if they lie in the same plane. Since they all pass through the origin $(0, 0, 0)$,they are coplanar if and only if the determinant of their direction cosines is zero.This is because the lines are coplanar if there exists a non-zero normal vector $(l, m, n)$ such that it is perpendicular to all three lines.Thus,we have the system of equations:$l l_1 + m m_1 + n n_1 = 0$$l l_2 + m m_2 + n n_2 = 0$$l l_3 + m m_3 + n n_3 = 0$For a non-trivial solution $(l, m, n) 
eq (0, 0, 0)$,the determinant of the coefficient matrix must be zero:$\left| \begin{array}{ccc} l_1 & m_1 & n_1 \ l_2 & m_2 & n_2 \ l_3 & m_3 & n_3 \end{array} ight| = 0$By swapping the second and third columns,the sign of the determinant changes,but it remains equal to zero:$-\left| \begin{array}{ccc} l_1 & n_1 & m_1 \ l_2 & n_2 & m_2 \ l_3 & n_3 & m_3 \end{array} ight| = 0$Therefore,$\left| \begin{array}{ccc} l_1 & n_1 & m_1 \ l_2 & n_2 & m_2 \ l_3 & n_3 & m_3 \end{array} ight| = 0$ is the correct condition.

The direction cosines of three lines passing through the origin are $(l_1, m_1, n_1)$,$(l_2, m_2, n_2)$,and $(l_3, m_3, n_3)$. The lines will be coplanar if

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