Show that if the axes are rectangular the equation of line through point `(x_1,y_1,z_1)` at right angle to the lines `x/l_1=y/m_1=z/n_1,x/l_2=y/m_2=z/n_2` is `(x-x_1)/(m_1n_2-m_2n_1)=(y-y_1)/(n_1l_2-n_2l_1)=(z-z_1)/(l_1m_2-l_2m_1)`

Find the equation of the plane through the line (x-x_1)/l_1=(y-y_1)/m_1=(z-z_1)/n_1 and parallel to the line (x-alpha)/l_2=(y-beta)/m_2=(z-gamma)/n_2

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

The length of projection of the segment join (x_1,y_1,z_1) and (x_2,y_2,z_20 on te line (x-alpha)/l=(y-beta)/m=(z-gamma)/n is (A) |l(x_2-x_1)+m(y_2-y_1)+n(z_2-z_1) (B) |alpha(x_2-x_1)+beta(y_2-y_1)+gamma(z_2-z_1)| (C) |(x_2-x_1)/l+(y_2-y_1)/m+(z_2-z_1)/n| (D) none of these

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

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

The equation of the plane containing the line (x-x_1)/l=(y-y_1)/m=(z-z_1)/n is a(x-x_1)+b(y-y_1)+c(z-z_1)=0,"" where (A) a x_1+b y_1+c z_1=0 (B) a l+b m+c n=0 (C) a/l=b/m=c/n (D) l x_1+m y_1+n z_1=0

Find the shortest distance between the skew-line4s l_1:(x-1)/2=(y+1)/1=(z-2)/4 and l_2 : (x+2)/4=(y-0)/(-3)=(z+1)/1

If (l_(1), m_(1), n_(1)) , (l_(2), m_(2), n_(2)) are D.C's of two lines, then (l_(1)m_(2)-l_(2)m_(1))^2+(m_(1)n_(2)-n_(1)m_(2))^2+(n_(1)l_(2)-n_(2)l_(1))^2+(l_(1)l_(2)+m_(1)m_(2)+n_(1)n_(2))^2=

If 1,m ,n are the direction cosines of the line of shortest distance between the lines (x-3)/(2) = (y+15)/(-7) = (z-9)/(5) and (x+1)/(2) = (y-1)/(1) = (z-9)/(-3) then :

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) .