If $\vec{a}$ and $\vec{b}$ are non-collinear vectors,then the value of $\alpha$ for which the vectors $\vec{u} = (\alpha - 2)\vec{a} + \vec{b}$ and $\vec{v} = (2 + 3\alpha)\vec{a} - 3\vec{b}$ are collinear is:

$A$ unit vector which is in the opposite direction of $(2, -2, 1)$ is $........$

Classify the following measure as a scalar or a vector:
$10^{-19} \text{ Coulomb}$

The position vector of the point which divides the line segment joining the points with position vectors $2a - 3b$ and $3a - 2b$ internally in the ratio $2 : 3$ is:

Let $ABCD$ be a parallelogram and $2\hat{i}+\hat{j}$,$4\hat{i}+5\hat{j}+4\hat{k}$ and $-\hat{i}-4\hat{j}-3\hat{k}$ be the position vectors of the vertices $A$,$B$,and $D$ respectively. Then the position vector of one of the points of trisection of the diagonal $AC$ is

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 $AG^2+BG^2+CG^2=$