$ABC$ is an equilateral triangle. The length of each side is $a$ and the centroid is point $O$. If $|\overrightarrow{AB} + \overrightarrow{BC} + \overrightarrow{AC}| = na$,find the value of $n$.

Let $\vec{A}$ and $\vec{B}$ be two non-zero vectors of different magnitudes. Which one of the following is the correct equation?

$A$ vector $P$ has $X$ and $Y$ components of magnitude $2$ units and $4$ units respectively. $A$ vector $Q$ along the negative $X$-axis has a magnitude of $6$ units. The vector $(Q - P)$ will be

Given that $A = B = C$. If $\vec A + \vec B = \vec C$,then the angle between $\vec A$ and $\vec C$ is $\theta_1$. If $\vec A + \vec B + \vec C = 0$,then the angle between $\vec A$ and $\vec C$ is $\theta_2$. What is the relation between $\theta_1$ and $\theta_2$?

If $r_1=2 \hat{x}$ and $r_2=2 \hat{y}$, where $\hat{x}$ and $\hat{y}$ are unit vectors along the $X$-axis and $Y$-axis respectively, then the magnitude of $r_1+r_2$ is

If the magnitude of the sum of two vectors is equal to the magnitude of the difference of the two vectors,the angle between these vectors is ........ $^o$.