If the position vectors of the vertices of a triangle are $a, b, c$,then find the sum of the vectors from the vertices to the centroid.

Let $O$ be the origin, $\vec{OP} = \vec{a}$ and $\vec{OQ} = \vec{b}$. If $R$ is the point on $\vec{OP}$ such that $\vec{OP} = 5\vec{OR}$, and $M$ is the point such that $\vec{OQ} = 5\vec{RM}$, then $\vec{PM}$ is equal to:

Answer the following as true or false.
$\vec{a}$ and $-\vec{a}$ are collinear.

Let $u, v, w$ be vectors such that $u + v + w = 0$. If $|u| = 3, |v| = 4,$ and $|w| = 5,$ find the value of $u \cdot v + v \cdot w + w \cdot u$.

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

If $\vec{a}, \vec{b}, \vec{c}$ are the position vectors of points $A, B, C$ respectively,and $D$ is the midpoint of $BC$,then $\vec{AD} = \dots$