Andhra Pradesh Board Solutions for Chapter: Direction Cosines and Direction Ratios, Exercise 2: Exercise 6(b)

Find the angle between the lines whose direction cosines satisfy the equations $l+m+n=0, l^2+m^2-n^2=0$.

If a ray makes angles $\alpha , \beta , \gamma , \delta$ with the four diagonals of a cube find ${\cos }^2\alpha +{\cos }^2\beta +{\cos }^2\gamma +{\cos }^2\delta$.

If $\left(l_1,m_1,n_1\right), \left(l_2,m_2,n_2\right)$ are direction cosines of two intersecting lines, show that direction cosines of two lines bisecting the angles between them are proportional to $\left(l_1\pm l_2,m_1\pm m_2,n_1\pm n_2\right)$.

$A(-1,2,-3), B(5,0,-6), C(0,4,-1)$ are three points. Show that the direction cosines of the bisectors of $\angle BAC$ are proportional to $(25,8,5)$ and $(-11,20,23)$.

If $(6,10,10), (1,0,-5), (6,-10,0)$ are vertices of a triangle, find the direction ratios of its sides. Determine whether it is right angled or isosceles.

The vertices of a triangle are $A(1,4,2), B(-2,1,2), C(2,3,-4).$ Find $\angle A, \angle B, \angle C$.

Find the angle between the lines whose direction cosines are given by the equations $3l+m+5n=0$ and $6mn-2nl+5lm=0$.

If a variable line in two adjacent positions has direction cosines $(l, m, n)$ and $(l+\delta l,m+\delta m,n+\delta n)$, show that the small angle $\delta \theta$ between the two positions is given by $(\delta \theta {)}^2=(\delta l{)}^2+(\delta m{)}^2+(\delta n{)}^2$.