If l_1, m_1, n_1 and l_2, m_2, n_2 are the di | vedclass

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(A) Let the direction vectors of the two perpendicular lines be $\vec{v_1} = (l_1, m_1, n_1)$ and $\vec{v_2} = (l_2, m_2, n_2)$.$A$ line perpendicular

Vedclass · Accepted Answer

$(m_1n_2 - m_2n_1), (n_1l_2 - n_2l_1), (l_1m_2 - l_2m_1)$

Vedclass · Answer

(A) Let the direction vectors of the two perpendicular lines be $\vec{v_1} = (l_1, m_1, n_1)$ and $\vec{v_2} = (l_2, m_2, n_2)$.$A$ line perpendicular to both lines will have a direction vector parallel to the cross product of the direction vectors of the two lines.The cross product $\vec{v_1} 	imes \vec{v_2}$ is given by the determinant:$\begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \ l_1 & m_1 & n_1 \ l_2 & m_2 & n_2 \end{vmatrix} = \hat{i}(m_1n_2 - m_2n_1) - \hat{j}(l_1n_2 - l_2n_1) + \hat{k}(l_1m_2 - l_2m_1)$.This simplifies to $\hat{i}(m_1n_2 - m_2n_1) + \hat{j}(n_1l_2 - n_2l_1) + \hat{k}(l_1m_2 - l_2m_1)$.Thus,the direction ratios of the line perpendicular to both are $(m_1n_2 - m_2n_1), (n_1l_2 - n_2l_1), (l_1m_2 - l_2m_1)$.Since the original lines are perpendicular and their direction vectors are unit vectors,the cross product results in a vector whose components represent the direction cosines of the common perpendicular line.

If $l_1, m_1, n_1$ and $l_2, m_2, n_2$ are the direction cosines of two perpendicular lines,then the direction cosines of the line which is perpendicular to both the lines will be:

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