If $O$ is the circumcentre and $O'$ is the orthocentre of the triangle $ABC$,then $\overrightarrow{O'A} + \overrightarrow{O'B} + \overrightarrow{O'C} = $

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

If $\overline{OA} = 3\hat{i} + \hat{j} - \hat{k}$,$|\overline{AB}| = 2\sqrt{6}$ and the direction ratios of $\overline{AB}$ are $1, -1, 2$,then $|\overline{OB}| = $

If $ABCDEF$ is a regular hexagon and $\vec{AB} + \vec{AC} + \vec{AD} + \vec{AE} + \vec{AF} = p\vec{AD} = q\vec{AO}$, where $O$ is the center of the hexagon, then the values of $p$ and $q$ respectively are

Let $\overline{OA}=\overline{a}, \overline{OB}=\overline{b}$. If the vector along the angle bisector of $\angle AOB$ is given by $x \frac{\overline{a}}{|\overline{a}|}+y \frac{\overline{b}}{|\overline{b}|}$,then which of the following is true?

The position vectors of the points $A, B, C$ are $(2\hat{i}+\hat{j}-\hat{k}), (3\hat{i}-2\hat{j}+\hat{k})$ and $(\hat{i}+4\hat{j}-3\hat{k})$ respectively. These points