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

The resultant of $\vec A$ and $\vec B$ makes an angle $\alpha $ with $\vec A$ and $\beta $ with $\vec B$,

When vector $\overrightarrow{ A }=2 \hat{ i }+3 \hat{ j }+2 \hat{ k }$ is subtracted from vector $\vec{B}$, it gives a vector equal to $2 \hat{j}$. Then the magnitude of vector $\vec{B}$ will be:

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

The resultant of two forces, one double the other in magnitude, is perpendicular to the smaller of the two forces. The angle between the two forces is ........ $^o$

A body is moving under the action of two forces ${\vec F_1} = 2\hat i - 5\hat j\,;\,{\vec F_2} = 3\hat i - 4\hat j$. Its velocity will become uniform under an additional third force ${\vec F_3}$ given by