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 $l+3 m+2 n=$

The point of intersection of the line passing through the point $\hat{\mathbf{i}}-\hat{\mathbf{j}}, \hat{\mathbf{j}}-\hat{\mathbf{k}}$ and the plane passing through the points $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}, 2 \hat{\mathbf{j}}-\hat{\mathbf{k}}, \hat{\mathbf{i}}+2 \hat{\mathbf{k}}$ is

If the vectors $\mathbf{B C}=2 \hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{C D}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ represent two adjacent sides of a parallelogram ABCD and $\theta$ is the angle between its diagonals $\mathbf{A C}$ and $\mathbf{B D}$, then $\tan \theta=$

A plane $\pi$ passing through the point $3 \hat{\mathbf{i}}-4 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$ is parallel to the plane which passes through the point $\hat{\mathbf{i}}+\hat{\mathbf{j}}-\hat{\mathbf{k}}$ and perpendicular to the vector $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$. Then, the cartesian equation of $\pi$ is