Let $G$ be the centroid of a triangle $ABC$ and $O$ be any other point in that plane,then $\overline{OA}+\overline{OB}+\overline{OC}+\overline{OG}=$

If $x\vec{a} + y\vec{b} + z\vec{c} = \vec{0}$,then under what condition are the three points $A, B, C$ with position vectors $\vec{a}, \vec{b}, \vec{c}$ collinear?

The position vector of point $C$ relative to $B$ is $(\hat{i} + \hat{j})$ and the position vector of $B$ relative to $A$ is $(\hat{i} - \hat{j})$. The position vector of $C$ relative to $A$ is:

If $\overline{a} = \bar{i} - 2\bar{j} + 2\bar{k}$ and $\overline{b} = 9\bar{i} + 6\bar{j} - 18\bar{k}$ are two vectors,then $\frac{\text{Projection of } \overline{b} \text{ on } \overline{a}}{\text{Projection of } \overline{a} \text{ on } \overline{b}} = $

The unit vector in the direction of the vector $\vec{a} = (2, 2, -1)$ is $......$

If $a, b, c$ are three non-coplanar vectors such that $a + b + c = \alpha d$ and $b + c + d = \beta a$,then $a + b + c + d$ is equal to