What is the minimum number of coplanar vectors having different magnitudes that can be added to give a zero resultant?

Two vectors of magnitude $3$ and $4$ have a resultant which makes angles $\alpha$ and $\beta$ respectively with them (given $\alpha + \beta \neq 90^o$).

The ratio of the maximum and minimum magnitudes of the resultant of two vectors $\vec{a}$ and $\vec{b}$ is $3 : 1$. Then $|\vec{a}|$ is equal to:

$A$ vector $\overrightarrow{A}$ when added to the sum of the vectors $(\hat{\imath}-2 \hat{\jmath}+2 \hat{k})$ and $(-2 \hat{\imath}+\hat{\jmath}-\hat{k})$ gives a unit vector along the $y$-axis. The magnitude of the vector $\overrightarrow{A}$ is

$\overrightarrow{A}$ and $\overrightarrow{B}$ are two vectors of equal magnitudes and $\theta$ is the angle between them. The angle between $\overrightarrow{A}$ or $\overrightarrow{B}$ with their resultant is

If three vectors have equal magnitude,i.e.,$A = B = C$,then the angle between $\vec{A}$ and $\vec{C}$ is $\alpha$. If $\vec{A} + \vec{B} + \vec{C} = 0$,then the angle between $\vec{A}$ and $\vec{C}$ is $\beta$. Find the ratio $\frac{\alpha}{\beta}$.