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The direction cosine of a line which is $\perp$ to two perpendicular lines whose direction cosines are $(l_1,m_1,n_1)\:\:and\:\:(l_2,m_2,n_2)$ is ?

$\begin{array}{1 1} (a)\:(m_1n_2-m_2n_1,\:n_1l_2-n_2l_1\:,l_1,m_2-m_1l_2) \\ (b)\:(l_1+l_2,\:m_1+m_2,\:n_1+n_2) \\ (c)\:(l_1+l_2,\:m_1+m_2,\:n_1+n_2) \\ (d)\:None\:of\:these. \end{array}$

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Any line $\perp$ to given two lines will be along $\overrightarrow b_1\times\overrightarrow b_2$
where $\overrightarrow b_1\:\:and\:\:\overrightarrow b_2$ are $d.r.^s$ of the given two lines.
Here, $\overrightarrow b_1=(l_1,m_1,n_1)\:\:and\:\:\overrightarrow b_2=(l_2,m_2,n_2)$
$\therefore$ The $d.c.$ of the required line is $\overrightarrow b_1\times\overrightarrow b_2=(m_1n_2-m_2n_1,\:n_1l_2-l_1n_2,\:l_1m_2-l_2m_1)$
answered Jan 8, 2014

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