$\overrightarrow A = 2\hat i + \hat j,\,B = 3\hat j - \hat k$ and $\overrightarrow C = 6\hat i - 2\hat k$.Value of $\overrightarrow A - 2\overrightarrow B + 3\overrightarrow C $ would be

A particle is situated at the origin of a coordinate system. The following forces begin to act on the particle simultaneously (Assuming particle is initially at rest)

${\vec F_1} = 5\hat i - 5\hat j + 5\hat k$            ${\vec F_2} = 2\hat i + 8\hat j + 6\hat k$

${\vec F_3} =  - 6\hat i + 4\hat j - 7\hat k$         ${\vec F_4} =  - \hat i - 3\hat j - 2\hat k$

Then the particle will move

The vectors $5i + 8j$ and $2i + 7j$ are added. The magnitude of the sum of these vector is

The coordinates of a moving particle at any time $t$ are given by $x = a\, t^2$ and $y = b\, t^2$. The speed of the particle is

$100$  coplanner forces each equal to  $10\,\,N$  act on a body. Each force makes angle $\pi /50$ with the preceding force. What is the resultant of the forces.......... $N$

Two forces $\vec{F}_1$ and $\vec{F}_2$ are acting on a body. One force has magnitude thrice that of the other force and the resultant of the two forces is equal to the force of larger magnitude. The angle between $\vec{F}_1$ and $\overrightarrow{\mathrm{F}}_2$ is $\cos ^{-1}\left(\frac{1}{\mathrm{n}}\right)$. The value of $|\mathrm{n}|$ is__________.