Three forces $\vec{F_1} = 2\hat{i} - 3\hat{j}$, $\vec{F_2} = -5\hat{i} + \hat{j}$, and $\vec{F_3} = a\hat{i} + b\hat{j}$ act on a particle. If the particle is in equilibrium, the values of a and b are:

A plane flies due east at an airspeed of 100 km/h. A wind is blowing from the south at 20 km/h. What is the plane's ground speed?

A force of 20 N acts along the positive x-axis. What is the magnitude of its y-component?

If $0.5\hat i + 0.8\hat j + {\rm{c\hat k}}$ is a unit vector, then the value of c is

\mathop {{\rm{ }}A}\limits^ -   = \mathop {{\rm{ }}B}\limits^ -   + \mathop {{\rm{ }}C}\limits^ -   and the magnitudes of \mathop {{\rm{ }}A}\limits^ -  ,\mathop {{\rm{ }}B}\limits^ -  ,\mathop {{\rm{ }}C}\limits^ -   are 5, 4 and 3 units respectively, the angle between \mathop {{\rm{ }}A}\limits^ -  \mathop {{\rm{ }}C}\limits^ -   is:

If a vector $\vec{A}$ is resolved into two perpendicular components $A_x$ and $A_y$, and the magnitude of $A_x$ is twice the magnitude of $A_y$, what is the angle between $\vec{A}$ and the x-axis?

A force $\vec{F} = (3\hat{i} + 4\hat{j}) \, N$ acts on a particle. What is the magnitude of the component of this force along the direction $\hat{i} + \hat{j}$?

A projectile is launched with an initial velocity $v_0$ at an angle $heta$ with the horizontal. At the highest point of its trajectory, the ratio of its potential energy to its kinetic energy is:

Two vectors $\vec{A}$ and $\vec{B}$ have magnitudes 5 and 12 units, respectively, and the magnitude of their resultant is 13 units. The angle between the vectors is:

A child travelling in a train throws a ball outside with a speed V. According to a child who is standing on the ground, the speed of the ball is

A river flows due east with a speed of 3 m/s. A man can swim in still water at a speed of 5 m/s. If he wants to cross the river in the shortest possible time, the angle at which he should swim with respect to the north is: