If $G$ is the centroid of the $\triangle ABC$,then $\vec{GA} + \vec{GB} + \vec{GC}$ is equal to

What should be added to the vector $a = 3i + 4j - 2k$ to obtain the resultant vector $i$?

$A$ vector $\bar{a}$ has components $1$ and $2p$ with respect to a rectangular Cartesian system. This system is rotated through a certain angle about the origin in the counter-clockwise sense. If,with respect to the new system,$\bar{a}$ has components $1$ and $(p+1)$,then:

Let $u$ and $v$ be non-collinear vectors in $\mathbb{R}^2$. Let $w$ be the orthogonal projection vector of $u$ on $v$. Consider two statements:
$(i)$ Any vector in $\mathbb{R}^2$ can be written as a linear combination of $u$ and $v$.
(ii) $w$ can be written as a linear combination of $u$ and $v$ as $w = au + bv$,where both $a$ and $b$ are non-zero real numbers.

The sum of two forces is $18 \ N$ and the resultant,whose direction is at right angles to the smaller force,is $12 \ N$. The magnitudes of the two forces are:

If $G$ is the centroid of the $\triangle ABC$,then $\vec{GA} + \vec{GB} + \vec{GC}$ is equal to