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 prove that the line whose direction cosines are proportional to ${l_1} + {l_2} + {l_3},{m_1} + {m_2} + {m_3}$ and ${n_1} + {n_2} + {n_3}$ makes equal angles with them.

Let $\overrightarrow {\rm{a}} = l\widehat {\rm{i}} + {m_1}\widehat {\rm{j}} + {n_1}\widehat {\rm{k}}$

$\overrightarrow {\rm{b}} = l\widehat {\rm{i}} + {m_2}\widehat {\rm{j}} + {n_2}\widehat {\rm{k}}$

$\overrightarrow {\rm{c}} = {l_3}\widehat {\rm{i}} + {m_3}\widehat {\rm{j}} + {n_3}\widehat {\rm{k}}$

$\overrightarrow {\rm{d}} = \left( {{l_1} + {l_2} + {l_3}} \right)\widehat {\rm{i}} + \left( {{m_1} + {m_2} + {m_2}} \right)\widehat {\rm{j}} + \left( {{n_1} + {n_2} + {n_3}} \right)\widehat {\rm{k}}$

Also, let $\alpha ,\beta$ and $\gamma$ are the angles between $\overrightarrow {\rm{a}}$ and $\overrightarrow {\rm{d}} ,\overrightarrow {\rm{b}}$

and $\overrightarrow {\rm{d}} ,\overrightarrow {\rm{c}}$ and $\overrightarrow {\rm{d}}$ .$\therefore \cos \alpha = {l_1}\left( {{l_1} + {l_2} + {l_3}} \right) + {m_1}\left( {{m_1} + {m_2} + {m_3}} \right) + {n_1}\left( {{n_1} + {n_2} + {n_3}} \right)$

$= l_1^2 + 4{l_2} + {l_1}{l_3} + m_1^2 + {m_1}{m_2} + {m_1}{m_3} + n_1^2 + {n_1}{n_2} + {n_1}{n_3}$

$= \left( {l_1^2 + m_1^2 + n_1^2} \right) + \left( {{l_1}{l_2} + {l_1}{l_3} + {m_1}{m_2} + {m_1}{m_3} + {n_1}{n_2} + {n_1}{n_3}} \right)$

$= 1 + 0 = 1$

. and$\left. {{l_1} \bot {l_2},{l_1} \bot {l_3},{m_1} \bot {m_2},{m_1} \bot {m_3},{n_1} \bot {n_2},{n_1} \bot {n_3}} \right]$

Similarly, $\cos \beta = {l_2}\left( {{l_1} + {l_2} + {l_3}} \right) + {m_2}\left( {{m_1} + {m_2} + {m_3}} \right) + {n_2}\left( {{n_1} + {n_2} + {n_3}} \right)$

$= 1 + 0$ and $\cos \gamma = 1 + 0$

$\Rightarrow$ $\cos \alpha = \cos \beta = \cos \gamma$

$\Rightarrow$ $\alpha = \beta = \gamma$

So, the line whose direction cosines are proportional to ${l_1} + {l_2} + {l_{3,}}{m_1} + {m_2} + {m_3}$, ${n_1} + {n_2} + {n_3}$

makes equal angles with the three mutually perpendicular lines whose direction cosines

are ${l_1},{m_1},{n_1},{l_2},{m_2},{n_2}$ and ${l_3},{m_3},{n_3}$ respectively.

OBJECTIVE TPYE QUESTIONS