For the given figure,which of the following relations is correct?

The magnitudes of two vectors are $A$ and $B$ $(A > B)$. If the maximum resultant magnitude of the two vectors is $n$ times their minimum resultant magnitude,then $\frac{A}{B} =$

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 $\theta_2$?

Let $\overrightarrow C = \overrightarrow A + \overrightarrow B$. Which of the following statements are correct?
$(A)$ It is possible to have $|\overrightarrow C| < |\overrightarrow A|$ and $|\overrightarrow C| < |\overrightarrow B|$.
$(B)$ $|\overrightarrow C|$ is always greater than $|\overrightarrow A|$.
$(C)$ $|\overrightarrow C|$ may be equal to $|\overrightarrow A| + |\overrightarrow B|$.
$(D)$ $|\overrightarrow C|$ is never equal to $|\overrightarrow A| + |\overrightarrow B|$.

$A$ car travels $6 \, km$ at an angle of $45^o$ north of east and then travels a distance of $4 \, km$ at an angle of $135^o$ north of east. How far is the final point from the starting point? What angle does the straight line joining its initial and final position make with the east direction?

If for two vectors $\overrightarrow{A}$ and $\overrightarrow{B}$,the sum $(\overrightarrow{A} + \overrightarrow{B})$ is perpendicular to the difference $(\overrightarrow{A} - \overrightarrow{B})$,then the ratio of their magnitudes is: