If l1, m1, n1 and l2, m2, n2 are the direction cosines of two concurrent lines, then the direction cosines of the lines bisecting the angles between them are proportional to :

Concept Used:

Direction Ratio is defined by cos θ where θ is the angle between the line and the Respective coordinate axis.

Calculation:

Let direction cosines of \(\mathrm{OX}=\left(\mathrm{l}_1, \mathrm{~m}_1, \mathrm{n}_1\right)\) and direction cosines of  \(\mathrm{OY}=\left(\mathrm{l}_2, \mathrm{~m}_2, \mathrm{n}_2\right)\)

Taking two points on OX and OY such that \(\mathrm{OX}=\mathrm{OY}=\mathrm{r}\)

Let the mid-point of XY be Z. Then, OZ is the bisector of the angle XOY.

Now, the coordinates of X and coordinates of Y are \(\left(l_2 r, m_2 r, n_2 r\right)\)
∴ The coordinates of Z are \(\frac{\left(\mathrm{l}_1+\mathrm{l}_2\right) \mathrm{r}}{2}, \frac{\left(\mathrm{m}_1+\mathrm{m}_2\right) \mathrm{r}}{2}, \frac{\left(\mathrm{n}_1+\mathrm{n}_2\right) \mathrm{r}}{2}\)

Hence, the direction cosines of OZ are proportional to \(\mathrm{l}_1+\mathrm{l}_2, \mathrm{~m}_1+\mathrm{m}_2\) and \(\mathrm{n}_1+\mathrm{n}_2\).