In $\triangle A B C$, if $a: b: c=4: 5: 6$, then the ratio of the circumradius to its inradius is

The perimeter of a $\triangle A B C$ is 6 times the arithmetic mean of the values of the sine of its angles. If its side $B C$ is of unit length, then $\angle A=$

If $|\mathbf{a}|=4,|\mathbf{b}|=5$ and $|\mathbf{a}-\mathbf{b}|=3$ and $\theta$ is the angle between the vectors $\mathbf{a}$ and $\mathbf{b}$, then $\cot ^2 \theta=$

If $A(1,2,3), B(3,7,-2), C(6,7,7)$ and $D(-1,0,-1)$ are points in a plane, then the vector equation of the line passing through the centroids of $\triangle A B D$ and $\triangle A C D$ is