If $l, m$ and $n$ are the d.c.'s of a normal to the plane passing through the points $(0,1,2)$, $(3,0,2)$ and $(4,5,0)$, then $|I|+|m|+|n|=$

Let $L$ be a line passing through the points $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+8 \hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+6 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$. Let $P$ be a plane passing through $-5 \hat{\mathbf{i}}+19 \hat{\mathbf{j}}-14 \hat{\mathbf{k}}$ and parallel to the vectors $\hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$. If $L$ meets the plane $P$ at a point $A$, then the position vector of $A$, is

If $\mathbf{r} \cdot(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}})=5, \mathbf{r} \cdot(\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}})=7$ are two planes and $(16,-9,0)$ is a point common to both the planes, then the vector equation of the line of intersection of the planes is $\mathbf{r}=$

$A(1,1,1), B(1,-4,3), C(2,-2,0)$ and $D(8,1,4)$ are the vertices of a tetrahedron. $G_1, G_2, G_3$ and $G_4$ are the centroids of the faces $A B C, B C D, C D A$ and $D A B$. Then, the centroid of the tetrahedron having $G_1, G_2, G_3$ and $G_4$ as its vertices is