# If $l_1,m_1,n_1,l_2,m_2,n_2,l_3,m_3,n_3$ are direction cosines of three mutually perpendicular lines,prove that the line whose direction cosines are proportional to $l_1+l_2+l_3,m_1+m_2+m_3,n_1+n_2+n_3$ make equal angles with them.

Toolbox:
• d.c is proportional to (a, b,c) means d.r = (a, b, c)
• To prove angle between a line with 3 different lines equal then prove $cos\theta_1=cos\theta_2=cos\theta_3$
• where $cos\theta_1=\bigg| \large\frac{d_1.d_2}{|d_1||d_2|} \bigg|$
Step 1
$d_1=(l_1, m_1, n_1)$
$d_2=(l_2, m_2, n_2)$ and $d_3=(l_3, m_3, n_3)$
$d_4 = ( l_1+l_2+l_3, \: m_1+m_2+m_3, \: n_1+n_2+n_3$
(say) $d_4 = ( a, \: \: \: \: \: \: \: \: \: \: \: \: \: \: b, \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: c)$
Step 2
given $d_1.d_2=d_2.d_3=d_1.d_3=0\: and \: |d_1|=|d_2|=|d_3|=0$
$\Rightarrow l_1l_2+m_1m_2+n_1n_2=0$
$\Rightarrow l_2l_3+m_2m_3+n_2n_3=0$
$\Rightarrow l_1l_3+m_1m_3+n_1n_3=0$
Step 3
Let the < between $d_4\: and \: d_1= \theta_1$
$d_4\: and \: d_2= \theta_2$
$d_4\: and \: d_3= \theta_3$
then $cos\theta_1 = \Large\frac{d_1.d_4}{|d_1||d_4|}$
$= \Large\frac{l_1a+m_1b+n_1c}{|d_4|}$
$= \Large\frac{l_1^2+m^2_1+n^2_1}{|d_4|}$
Step 4
Similarly find $cos\theta_2\: and \: cos\theta_3$ to be equal to $cos\theta_1$
Step 5
$cos\theta_1 = cos\theta_2=cos\theta_3$

edited Apr 5, 2013