If $\vec{A} + \vec{B} = \vec{C}$ and $A + B = C$,what is the angle between $\vec{A}$ and $\vec{B}$?

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:

The sum of the magnitudes of two vectors $\vec{A}$ and $\vec{B}$ is $8$,and the magnitude of their resultant is $4$. If the resultant vector is perpendicular to one of the vectors,then the magnitudes of the two vectors $\vec{A}$ and $\vec{B}$ are:

The resultant of two vectors $\vec{P}$ and $\vec{Q}$ is $\vec{R}$. When the direction of $\vec{Q}$ is reversed,the resultant is given by $\vec{S}$. Which one of the following is true for vectors $\vec{R}$ and $\vec{S}$?

Two forces are such that the sum of their magnitudes is $18 \; N$ and their resultant is $12 \; N$,which is perpendicular to the smaller force. Then the magnitudes of the forces are:

Two vectors having equal magnitudes of $x$ units acting at an angle of $45^\circ$ have a resultant of $\sqrt{2 + \sqrt{2}}$ units. The value of $x$ is