Let $\mathbf{a}=\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \mathbf{b}=6 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ and $\mathbf{c}=3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}-12 \hat{\mathbf{k}}$ be three vectors. If $\mathbf{p}$ is the projection of $\mathbf{b}$ on $\mathbf{a}$ and $\mathbf{q}$ is the projection of $\mathbf{c}$ on $\mathbf{a}$, then $13 \mathbf{p}=$

Let $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \mathbf{b}=3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ and $\mathbf{c}=\hat{\mathbf{i}}-4 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ be three vectors. Let $\mathbf{r}$ be a vector perpendicular to both $\mathbf{b}$, $c$ and $\mathbf{r} \cdot \mathbf{a}=11$. Then, the vector among the following that is perpendicular to $\mathbf{r}$ is

The volume of the tetrahedron with $\hat{\mathbf{i}}-\lambda \hat{\mathbf{j}}+\hat{\mathbf{k}}$, $\lambda \hat{\mathbf{i}}-\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\lambda \hat{\mathbf{k}}$ as coterminous edges is 2 . If $\lambda$ is an integer, then $|\lambda \hat{\mathbf{i}}-3 \lambda \hat{\mathbf{j}}+3 \hat{\mathbf{k}}|=$

Let $\mathbf{O A}=\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{O B}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{O C}=4 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ be the position vectors of three points, $A, B$ and $C$. Let $P$ be the point which divides $A B$ in the ratio $2: 1$. If $l, m, n$ are the direction cosines of the vector $\mathbf{P C}$, then $l+3 m+2 n=$