Let `l_1,m_1,n_1; l_2,m_2,n_2` and `l_3,m_3,n_3` be the direction cosines of three mutually perpendicular lines. Show that the direction ratios of the line which makes equal angles with each of them are `(l_1+l_2+l_3),(m_1+m_2+m_3),(n_1+n_2+n_3)`

If l_1,""""m_1,""""n_1 and l_2,""""m_2,""""n_2 are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m_1n_2-m_2n_1, n_1l_2-n_2l_1, l_1m_2-l_2m_1 .

If l_(1), m_(1), n_(1) and l_(2),m_(2),n_(2) are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m_(1)n_(2)-m_(2)n_(1),n_(1)l_(2)-n_(2)l_(1),l_(1)m_(2)-l_(2)m_(1) .

If l_(1), m_(1), n_(1) and l_(2),m_(2),n_(2) are the direction cosines of two mutually perpendicular lines, show that the direction cosines of the line perpendicular to both of these are m_(1)n_(2)-m_(2)n_(1),n_(1)l_(2)-n_(2)l_(1),l_(1)m_(2)-l_(2)m_(1) .

If l_1, m_1, n_2 ; l_2, m_2, n_2, be the direction cosines of two concurrent lines, then direction cosines of the line bisecting the angles between them are proportional to

Let l_i,m_i,n_i; i=1,2,3 be the direction cosines of three mutually perpendicular vectors in space. Show that AA'=I_3 where A=[[l_1,m_1,n_1] , [l_2,m_2,n_2] , [l_3,m_3,n_3]]

If l_(1), m_(1), n_(1), l_(2), m_(2), n_(2) and l_(3), m_(3), n_(3) are direction cosines of three mutuallyy perpendicular lines then, the value of |(l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)),(l_(3),m_(3),n_(3))| is

If ,"l"_1,"m"_1,("\ n")_1("\ and\ l")_2,"m"_2,"n"_2 be the direction cosines of two lines, show that the directioin cosines of the line perpendicular to both them are proportional to ("m"_1("\ n")_2-"m"_2"n"_1),"\ "("n"_1"l"_2-"n"_2"l"_1),"\ "("l"_1"m"_2-"l"_2"m"_1)

If l_1,m_1,n_1 and l_2,m_2,n_2 are the direction cosines of two rays OA and OB making an angle theta then show that the direction cosines of the bisector of /_AOB are (l_1+l_2)/(2cos(theta/2)),(m_1+m_2)/(2cos(theta/2)),(n_1+n_2)/(2cos(theta/2))

The direction cosines of a line equally inclined to three mutually perpendiclar lines having direction cosines as l_(1),m_(1),n_(1),l_(2),m_(2),n_(2) and l_(3), m_(3),n_(3) are

Find the direction cosines of the lines, connected by the relations: l+m+n=0 and 2l m+2ln-m n=0.