If the direction cosines of a variable line in two adjacent points be `l, m, n and l+deltal,m+deltam,n+deltan` the small angle `deltatheta`as between the two positions is given by

A variable line in two adjacent positions has direction cosines l, m, n and l+delta l, m+deltam, n+delta n . Prove that the small angle delta theta between the two positions of the variable line is given by, (delta theta)^(2)=(delta l)^(2)+(delta m)^(2)+(delta n)^(2)

A variable plane l x+m y+n z=p(w h e r el ,m ,n are direction cosines of normal ) intersects the coordinate axes at points A ,Ba n dC , respectively. Show that the foot of the normal on the plane from the origin is the orthocenter of triangle A B C and hence find the coordinate of the circumcentre of triangle A B Cdot

The coordinates of the mid-point of the line segment joining the points (l, 2m) and (-l +2m, 2l -2m) are

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

The direction cosines of a line satisfy the relations lamda(l+m)=nandmn+nl+lm=0 . The value of lamda for which the two lines are perpendicular to each other, is

The coordinates of the mid-point of the line segment obtained by joining the points (l, m) and (l + m, l - m) are

The direction cosines of two lines satisfy the relations lamda(l+m)=n and mn+nl+lm=0 . The value of lamda , for which the two lines are perpendicular to each other, is -

Find the direction cosines of the two lines which are connected by the relations l-5m+3n=0 and 7l^(2)+5m^(2)-3n^(2)=0-

Find the angle between the line whose direction cosines are given by l+m+n=0a n dl^2+m^2-n^2=0.

If l_(1), m_(1), n_(1) and l_(2), m_(2), n_(2) are direction cosines of two mutually perpendicular straight lines, then prove that the direction cosines of the straight line which is perpendicular to both the given lines are pm (m_(1)n_(2)-m_(2)n_(1)), pm (n_(1)l_(2)-n_(2)l_(1)), pm (l_(1)m_(2)-l_(2)m_(1))