In a $\triangle A B C, D$ and $E$ divide the sides $B C$ and $C A$ in the ratio $2: 1$ respectively. If $P$ is the point of intersection of $A D$ and $B E$, then the ratio in which $P$ divides $A D$ is

If the points with position vectors $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, 2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-4 \hat{\mathbf{k}},-3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ and $a \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$ are coplanar, then $a=$

If $P$ is a point on the line parallel to the vector $2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}-6 \hat{\mathbf{k}}$ and passing through the point $A$ whose position vector is $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $A P=21$, then the position vector of $P$ can be

Let $\mathbf{a}$ be a vector in the plane containing vectors $\mathbf{b}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\mathbf{c}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}$. If $\mathbf{a}$ is perpendicular to $\hat{\mathbf{i}}+\hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ and its projection on $\mathbf{b}$ is $3 \sqrt{6}$, then $|\mathbf{a}|^2=$