If $\bar{a}, \bar{b}, \bar{c}$ are non-zero vectors such that no two of them are parallel,and $\bar{a} + \bar{b}$ is parallel to $\bar{c}$,and $\bar{b} + \bar{c}$ is parallel to $\bar{a}$,then $\bar{a} + \bar{b} + \bar{c} = $

Let $u, v$ and $w$ be three vectors in $R^3$. Then,any vector $z \in R^3$ can be written as $z = au + bv + cw$ for some scalars $a, b$ and $c$ if and only if:

Let $\vec{a}, \vec{b}, \vec{c}$ be three non-coplanar vectors. Then the point of intersection of the line joining the points $\vec{a}+\vec{b}+\vec{c}, \vec{a}-\vec{b}+3 \vec{c}$ and the line joining the points $2 \vec{a}-\vec{b}+\vec{c}, \vec{a}-2 \vec{b}+4 \vec{c}$ is

If $\vec{PO} + \vec{OQ} = \vec{QO} + \vec{OR}$,then

Classify the following measure as a scalar or a vector:
$10 \text{ Newton}$

The perimeter of a triangle with sides represented by vectors $\vec{a} = 3\hat{i} + 4\hat{j} + 5\hat{k}$,$\vec{b} = 4\hat{i} - 3\hat{j} - 5\hat{k}$,and $\vec{c} = 7\hat{i} + \hat{j}$ is: