In a $\triangle A B C$, if $r_{1}=6, r_{2}=9, r_{3}=18$, then $\cos A=$

$2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+\hat{\mathbf{k}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}$ are the position vectores of two points $A$ and $B$ respectively and $C$ divides $A B$ in the ratio $3: 2$ : If $3 \hat{\mathbf{i}}-\hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is the position of vector of a point $D$, then the unit vector in the direction of $C D$ is

A plane $\pi$ passing through the points $2 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}, 3 \hat{\mathbf{i}}+4 \hat{\mathbf{k}}$ is parallel to the vector $2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-4 \hat{\mathbf{k}}$. If a line joining the points $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$ and $\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ intersects the plane $\pi$ at the point $a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$, then $a+b+2 c=$

A unit vector $\hat{\mathbf{e}}=a \hat{\mathbf{i}}+b \hat{\mathbf{j}}+c \hat{\mathbf{k}}$ is coplanar with the vectors $\hat{\mathbf{i}}-3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$, and $3 \hat{\mathbf{i}}+\hat{\mathbf{j}}-5 \hat{\mathbf{k}}$. If $\hat{\mathbf{e}}$ is perpendicular to the vector $\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}$, then $2 a^{2}+3 b^{2}+4 c^{2}=$