Let< l, m, n> be the direction cosines of the line, which is perpendicular to two lines whose direction cosines are : $\langle l_1, m_1, n_1 \rangle$ and $\langle l_2, m_2, n_2 \rangle$ . $\therefore ll_1 + mm_1 + nn_1 = 0$ ...(1) and $ll_2 + mm_2 + nn_2 = 0$ ...(2) Solving (1) and (2), by cross-multiplication, we get $=\frac{1}{\sin\theta} = \frac{1}{\sin 90^{\circ}} = 1$Hence $l = m_1n_2 - m_2n_1$, $m = n_1l_2 - n_2l_1$, $n = l_1m_2 - l_2m_1$.