In a $\triangle A B C, a=5, b=4$ and $\tan \frac{C}{2}=\sqrt{\frac{7}{9}}$, then its inradius $r=$

Two adjacent sides of a triangle are represented by the vectors $2 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $2 \sqrt{3} \hat{\mathbf{i}}-2 \sqrt{3} \hat{\mathbf{j}}+\sqrt{3} \hat{\mathbf{k}}$. Then, the least angle of the triangle and perimeter of the triangle are respectively.

A plane $\pi_1$ contains the vectors $\hat{\mathbf{i}}+\hat{\mathbf{j}}$ and $\hat{\mathbf{i}}+2 \hat{\mathbf{j}}$. Another plane $\pi_2$ contains the vectors $2 \hat{\mathbf{i}}-\hat{\mathbf{j}}$ and $3 \hat{\mathbf{i}}+2 \hat{\mathbf{k}}$. $\mathbf{a}$ is a vectors parallel to the line of intersection of $\pi_1$ and $\pi_2$. If the angle $\theta$ between $\mathbf{a}$ and $\hat{\mathbf{i}}-2 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}$ is acute, then $\theta=$

In a quadrilateral $A B C D, \mathbf{A}=\frac{2 \pi}{3}$ and $A C$ is the bisector of angle $\mathbf{A}$. If $15|\mathbf{A C}|=5|\mathbf{A D}|=3|\mathbf{A B}|$, then angle between $\mathbf{A B}$ and $\mathbf{B C}$ is