$ABC$ is an equilateral triangle. Length of each side is $a$ and centroid is point $O$. Find If $|\overrightarrow{A B}+\overrightarrow{B C}+\overrightarrow{A C}|=n a$ then $n =$ ?

The resultant of two vectors $\overrightarrow P $ and $\overrightarrow Q $ is $\overrightarrow R .$ If $Q$ is doubled, the new resultant is perpendicular to $P$. Then $R $ equals

A car moves towards north at a speed of $54 \,km / h$ for $1 \,h$. Then it moves eastward with same speed for same duration. The average speed and velocity of car for complete journey is ..........

$ABC$ is an equilateral triangle. Length of each side is $a$ and centroid is point $O$. Find $\overrightarrow{A B}+\overrightarrow{A C}=n \overrightarrow{A O}$ then $n =  ........  $

Statement $I :$Two forces $(\overrightarrow{{P}}+\overrightarrow{{Q}})$ and $(\overrightarrow{{P}}-\overrightarrow{{Q}})$ where $\overrightarrow{{P}} \perp \overrightarrow{{Q}}$, when act at an angle $\theta_{1}$ to each other, the magnitude of their resultant is $\sqrt{3\left({P}^{2}+{Q}^{2}\right)}$, when they act at an angle $\theta_{2}$, the magnitude of their resultant becomes $\sqrt{2\left({P}^{2}+{Q}^{2}\right)}$. This is possible only when $\theta_{1}<\theta_{2}$.

Statement $II :$ In the situation given above. $\theta_{1}=60^{\circ} \text { and } \theta_{2}=90^{\circ}$ In the light of the above statements, choose the most appropriate answer from the options given below

Given that $\overrightarrow A + \overrightarrow B = \overrightarrow C $and that $\overrightarrow C $ is $ \bot $ to $\overrightarrow A $. Further if $|\overrightarrow A |\, = \,|\overrightarrow C |,$then what is the angle between $\overrightarrow A $ and $\overrightarrow B $