Question

$A=\begin{bmatrix}l_{1}&m_{1}&n_{1}\\ l_{2}&m_{2}&n_{2}\\ l_{3}&m_{3}&n_{3}\end{bmatrix} and B= \begin{bmatrix}p_{1}&q_{1}&r_{1}\\ p_{2}&q_{2}&r_{2}\\ p_{3}&q_{3}&r_{3}\end{bmatrix}$ 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 $\left(l_{1}, m_{1}, n_{1}\right),\left(l_{2}, m_{2}, n_{2}\right)$ and $\left(l_{3}, m_{3}, n_{3}\right)$ are the direction cosines of three mutually perpendicular lines then $\left(p_{1}, q_{1}, r_{1}\right),\left(p_{2}, q_{2}, r_{2}\right)$ and $\left(p_{3}, q_{3}, r_{3}\right)$ are

• A. the direction cosines of three mutually perpendicular lines
• B. the direction ratios of three mutually perpendicular lines which are not direction cosines
• C. the direction cosines of three lines which need not be perpendicular
• D. the direction ratios but not the direction cosines of three lines which need not be perpendicular

Answer 1

Answer: (A)

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