If $R: r_1: r=5: 12: 2$, then $r+r_3+r_2-r_1=$

If $12 \hat{\mathbf{i}}-12 \hat{\mathbf{j}}-18 \hat{\mathbf{k}},-3 \hat{\mathbf{i}}-6 \hat{\mathbf{j}}-9 \hat{\mathbf{k}}$ and $3 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-24 \hat{\mathbf{k}}$ be the position vectors of the vertices $A, B$ and $C$ respectively of $\triangle A B C$, then the position vector of the incentre of $\triangle A B C$ is

Let $\Pi$ be a plane containing the points $(0,-5,-1),(1,-2,5),(-3,5,0)$ and $L$ be a line passing through the point $(0,-5,-1)$ and parallel to the vector $\hat{\mathbf{i}}+5 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}$. Then the length of the projection of the unit normal vector to the plane $\Pi$ on the line $L$ is

For non-coplanar vectors $\mathbf{a}, \mathbf{b}$ and $\mathbf{c}$, if the point of intersection of the line $\mathbf{r}=\mathbf{a}+t(\mathbf{b}-\mathbf{c})$ and the plane $\mathbf{r}=\mathbf{b}+\mathbf{c}+x(\mathbf{a}-\mathbf{b})+y(\mathbf{c}+\mathbf{a})$ is $l \mathbf{a}+m \mathbf{b}+n \mathbf{c}$, then $3 l+4 m+2 n=$