Mathematics - 3D Coordinate Geometry Question with Solution | TestHub

Take $OA=OB=1$, so $A\left(l_1,m_1,n_1\right)$ and $B\left(l_2,m_2,n_2\right)$. The bisector OC passes through the mid-point C of AB, $C\left(\frac{l_1+l_2}{2},\frac{m_1+m_2}{2},\frac{n_1+n_2}{2}\right)$. Then $OC=\frac{1}{2}\sqrt{2+2\cos \theta }=\cos \frac{\theta }{2}$, using $\cos \theta =l_1{l}_2+m_1{m}_2+n_1{n}_2$. So the d.c.'s of OC are $\frac{l_1+l_2}{2\cos \theta /2},\frac{m_1+m_2}{2\cos \theta /2},\frac{n_1+n_2}{2\cos \theta /2}$.