If $A(1,2,3), B(2,3,-1), C(3,-1,-2)$ are the vertices of a triangle $ABC$,then the direction ratios of the internal angle bisector of $\angle A$ are:

Consider a line $L$ passing through the points $P(1, 2, 1)$ and $Q(2, 1, -1)$. If the mirror image of the point $A(2, 2, 2)$ in the line $L$ is $(\alpha, \beta, \gamma)$,then $\alpha + \beta + 6\gamma$ is equal to:

The angle between the lines $\frac{x + 4}{1} = \frac{y - 3}{2} = \frac{z + 2}{3}$ and $\frac{x}{3} = \frac{y - 1}{-2} = \frac{z}{1}$ is

If the line passes through the points $P(6, -1, 2)$,$Q(8, -7, 2\lambda)$,and $R(5, 2, 4)$,then the value of $\lambda$ is $.......$

Prove that the lines $x=p y+q, z=r y+s$ and $x=p^{\prime} y+q^{\prime}, z=r^{\prime} y+s^{\prime}$ are perpendicular if $p p^{\prime}+r r^{\prime}+1=0$.

If lines $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-k}{2}=\frac{z-0}{1}$ intersect,then the value of $k$ is