For any two vectors $\vec{a}$ and $\vec{b}$,which of the following statements is true?

Consider the following statements :
Statement $(I)$ : If either $|\vec{a}|=0$ or $|\vec{b}|=0$,then $\vec{a} \cdot \vec{b}=0$.
Statement $(II)$ : If $\vec{a} \times \vec{b}=\vec{0}$,then $\vec{a}$ is perpendicular to $\vec{b}$.
Which of the following is correct?

If $a, b, c$ are the position vectors of the vertices $A, B, C$ of the triangle $ABC,$ then the centroid of $\Delta ABC$ is

If $\vec{a}, \vec{b}, \vec{c}$ are three vectors such that $|\vec{a}|=|\vec{b}|=|\vec{c}|=\sqrt{3}$ and $(\vec{a}+\vec{b}+\vec{c})^2+(\vec{b}+\vec{c}-\vec{a})^2+(\vec{c}+\vec{a}-\vec{b})^2=36$,then $|2 \vec{a}-3 \vec{b}+2 \vec{c}|^2=$

$(3, 0, 2)$ and $(0, 2, k)$ are the direction ratios of two lines and $\theta$ is the angle between them. If $|\cos \theta| = \frac{6}{13}$, then $k =$

If $\hat{i}$ is the position vector of the centroid $G$ of triangle $ABC$ and $2\hat{i}+\hat{j}+\hat{k}$ and $2\hat{i}+4\hat{j}-4\hat{k}$ are respectively the position vectors of its vertices $A$ and $B$,then $AG^2+BG^2+CG^2=$