If $A(4,7,8), B(2,3,4)$ and $C(2,5,7)$ are the vertices of $\triangle A B C$, then the length of the internal bisector of the angle $A$ is

For scalars $\lambda, \mu$ if the vector equation of a plane is $\mathbf{r}=(2+3 \lambda-\mu) \hat{\mathbf{i}}+(1-2 \lambda+3 \mu) \hat{\mathbf{j}}+(-2+2 \lambda+\mu) \hat{\mathbf{k}}$, then its Cartesian equation is

The position vectors of the points $A$ and $B$ are respectively $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ and $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$. If the points $P$ and $Q$ are respectively the orthogonal projections of $A$ and $B$ on the plane $x+y+z=3$, then $P Q=$

If $A(4,3,2), B(5,4,6), C(-1,-1,5)$ are the vertices of a triangle, then the coordinates of the point in which the bisector of the angle $A$ meet the side $B C$ is