Question

$A=\left[\begin{array}{ccc}{l}_1& m_1& n_1\\ l_2& m_2& n_2\\ l_3& m_3& n_3\end{array}\right]andB=\left[\begin{array}{ccc}{p}_1& q_1& r_1\\ p_2& q_2& r_2\\ p_3& q_3& r_3\end{array}\right]$ 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