If $|\vec{a}|=2$ and $|\vec{b}|=3$,find the value of $|3 \vec{a}+2 \vec{b}|$ based on the given figure.

Find the resultant of three vectors $\overrightarrow{OA}$,$\overrightarrow{OB}$,and $\overrightarrow{OC}$ as shown in the figure. The radius of the circle is $R$.

Resultant of two vectors $\vec{P}$ and $\vec{Q}$ is of magnitude $A$. If $\vec{Q}$ is reversed, then the resultant is of magnitude $B$. The value of $A^2 + B^2$ is

The magnitude of the sum of the two vectors $\vec{A}$ and $\vec{B}$ is equal to the magnitude of the difference of the two vectors $\vec{A}$ and $\vec{B}$. The angle between $\vec{A}$ and $\vec{B}$ is: (in $^{\circ}$)

The resultant of two vectors $\vec{P}$ and $\vec{Q}$ is $\vec{R}$. If $\vec{Q}$ is doubled,the new resultant vector is perpendicular to $\vec{P}$. What is the magnitude of $\vec{R}$?

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: