Prove that the three lines from O with direction cosines `l_1, m_1, n_1: l_2, m_2, n_2: l_3, m_3, n_3` are coplanar, if `l_1(m_2n_3-n_2m_3)+m_1(n_2l_3-l_2n_3)+n_1(l_2m_3-l_3m_2)=0`

Prove that the three lines from the origin O, with direction cosines l_1,m_1,n_1; l_2,m_2,n_2;l_3,m_3,n_3 are coplaner if |[l_1,m_1,n_1],[l_2,m_2,n_2],[l_3,m_3,n_3]|=0

Two lines with direction cosines l_(1),m_(1),n_(1) and l_(2), m_(2), n_(2) are at right angle of

The direction ratios of the bisector of the angle between the lines whose direction cosines are l_1,m_1,n_1 and l_2,m_2,n_2 are (A) l_1+l_2,m_1+m_2+n_1+n_2 (B) l_1-l_2,m_1-m_2-n_1-n_2 (C) l_1m_2-l_2m_1,m_1n_2-m_2n_1,n_1l_2-n_2l_1 (D) l_1m_2+l_2m_1,m_1n_2+m_2n_1,n_1l_2+n_2l_1

Two lines with direction cosines l_1,m_1,n_1 and l_2,m_2,n_2 are at righat angles iff (A) l_1l_2+m_1m_2+n_1n_2=0 (B) l_1=l_2,m_1=m_2,n_1=n_2 (C) l_1/l_2=m_1/m_2=n_1/n_2 (D) l_1l_2=m_1m_2=n_1n_2

The direction cosines of a line bisecting the angle between two perpendicular lines whose direction cosines are l_1,m_1,n_1 and l_2,m_2,n_2 are (1)(l_1+l_2)/2,(m_1+m_2)/2,(n_1+n_2)/2 (2)l_1+l_2,m_1+m_2,n_1+n_2 (3)(l_1+l_2)/(sqrt(2)),(m_1-m_2)/2,(n_1+n_2)/(sqrt(2)) (4)l_1-l_2,m_1-m_2,n_1-n_2 (5)"n o n eo ft h e s e"

If (l,m,n) are the direction cosines of a line then l^(2)+m^(2)+n^(2)=

If three mutually perpendicular lines have direction cosines (l_(1),m_(1),n_(1)),(l_(2),m_(2),n_(2)) and (L_(3),m_(3),n_(3)) then the line having direction cosines l_(1)+l_(2)+l_(3),m_(1)+m_(2)+m_(3), and n_(1)+n_(2)+n_(3) ,make an angle of

Statement 1: The direction cosines of one of the angular bisectors of two intersecting line having direction cosines as l_1,m_1, n_1a n dl_2, m_2, n_2 are proportional to l_1+l_2,m_1+m_2, n_1+n_2dot Statement 2: The angle between the two intersection lines having direction cosines as l_1,m_1, n_1a n dl_2, m_2, n_2 is given by costheta=l_1l_2+m_1m_2+n_1n_2dot

Show that the matris [[l_(1),m_(1),n_(1)],[l_(2),m_(2),n_(2)],[l_(3),m_(3),n_(3)]] is orthogonal, if l_(1)^(2) + m_(1)^(2) + n_(1)^(2) = Sigmal_(1)^(2) = 1 = Sigma l_(2)^(2) = Sigma_(3) ^(2) and l_(1) l_(2) + m_(1)m_(2) + n_(1) n_(2) = Sigma l_(1)l_(2) =0 = Sigma l_(2)l_(3) = Sigma l_(3) l_(1).