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$A = [\begin{array}{lll} l_{1} m_{1} n_{1} \\ l_{2} m_{2} n_{2} \\ l_{3} m_{3} n_{3} \end{array}] and B = [\begin{array}{ccc} p_{1} & q_{1} & r_{1} \\ p_{2} & q_{2} & r_{2} \\ p_{3} & q_{3} & r_{3} \end{array}] where p_{i}, q_{i}$ $r_{i} are the co-factors of the elements l_{i}, m_{i}, n_{i} for i = 1, 2, 3$
$If (l_{1}, m_{1}, n_{1}), (l_{2}, m_{2}, n_{2}) and (l_{3}, m_{3}, n_{3}) are the direction cosines of three mutually perpendicular lines then$
$(p_{1}, q_{1}, r_{1}), (p_{2}, q_{2}, r_{2}) and (p_{3}, q_{3}, r_{3}) are$

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the direction cosines of three mutually perpendicular lines

the direction ratios of three mutually perpendicular lines which are not direction cosines

the direction cosines of three lines which need not be perpendicular

the direction ratios but not the direction cosines of three lines which need not be perpendicular

answer is C.

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Detailed Solution

$Let \vec{a} = l_{1} \hat{i} + m_{1} \hat{j} + n_{1} \hat{k}, \vec{b} = l_{2} \hat{i} + m_{2} \hat{j} + n_{2} \hat{k} and$ $\vec{c} = l_{3} \hat{i} + m_{3} \hat{j} + n_{3} \hat{k}$

$Given that \vec{a}, \vec{b}, \vec{c} are three mutually perpendicular unit vectors$

$Then p_{1} \hat{i} + q_{1} \hat{j} + r_{1} \hat{k} = \vec{b} \times \vec{c} = \vec{a}$

$(∵ \vec{b} \times \vec{c} parallel to \vec{a} and \vec{b} \times \vec{c}, \vec{a} are unit vectors)$

$Similarly, p_{2} \hat{i} + q_{2} \hat{j} + r_{2} \hat{k} = \vec{c} \times \vec{a} = \vec{b}$

$and p_{3} \hat{i} + q_{3} \hat{j} + r_{3} \hat{k} = \vec{a} \times \vec{b} = \vec{c}$

These vectors are also mutually perpendicular unit vectors.

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