The position vector of a particle is $\vec{r} = (\hat{i} + 2\hat{j} - \hat{k})$ and its momentum is $\vec{p} = (3\hat{i} + 4\hat{j} - 2\hat{k})$. The angular momentum of this particle is perpendicular to:

$A$ cylinder of mass $5 \ kg$ and radius $30 \ cm$ is free to rotate about its axis. It receives an initial angular impulse of $3 \ kg \ m^2 s^{-1}$ and subsequently receives the same impulse every $4 \ s$. What will be the angular speed of the cylinder $30 \ s$ after the initial impulse? The cylinder is initially at rest.

$A$ particle of mass $5 \ kg$ moves along the straight line $y = 2x + 4$ with a velocity of $3\sqrt{5} \ m/s$. Find its angular momentum about the origin in $kg \ m^2 \ s^{-1}$.

$A$ particle of mass $0.5\, kg$ is rotating in a circular path of radius $2\, m$ and centripetal force on it is $9\, N$. Its angular momentum (in $J\cdot s$) is:

Two wheels are connected by a belt. The radius of the larger wheel is three times that of the smaller one. What is the ratio of the rotational inertia of the larger wheel to the smaller wheel,when both wheels have the same angular momentum?

$A$ body of mass $m$ is thrown with velocity $u$ from the origin of a coordinate system at an angle $\theta$ with the horizontal. The magnitude of the angular momentum of the particle about the origin at the time $t$ when it is at the maximum height of the trajectory is proportional to