The sum of the magnitudes of two forces is $25 \,N$. The resultant of these forces is perpendicular to the smaller force and has a magnitude of $10 \,N$. The two forces are:

$\overrightarrow{A}$ 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

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:

If $P + Q = R$ and $|P| = |Q| = \sqrt{3}$ and $|R| = 3$,then the angle between $P$ and $Q$ is

The vectors $(\vec{A} + \vec{B})$ and $(\vec{A} - \vec{B})$ are perpendicular to each other. This is possible under the condition:

The magnitudes of two vectors are $8$ units and $6$ units respectively. Find the magnitude of the resultant vector if the angle between these two vectors is $(i) \theta = 0^{\circ}$,$(ii) \theta = 180^{\circ}$,$(iii) \theta = 90^{\circ}$,and $(iv) \theta = 120^{\circ}$.