A rigid body is rotating with an angular velocity $\mathbf{\omega} = (2\mathbf{i} - 3\mathbf{j} + 4\mathbf{k}) \, \text{rad/s}$. A point A on the body is located at $\mathbf{r}_A = (1\mathbf{i} + 2\mathbf{j} - 1\mathbf{k}) \, \text{m}$ relative to a fixed origin O. Determine the velocity of point A if the body rotates about O.

A rigid body is rotating about a fixed point O with an angular velocity vector $\mathbf{\omega} = (2\hat{i} - 3\hat{j} + 4\hat{k}) \text{ rad/s}$ and an angular acceleration vector $\mathbf{\alpha} = (1\hat{i} + 2\hat{j} - 1\hat{k}) \text{ rad/s}^2$. Determine the velocity and acceleration of a point P on the body, given its position vector relative to O is $\mathbf{r} = (1\hat{i} + 1\hat{j} + 1\hat{k}) \text{ m}$.

The Coriolis effect is a quintessential example of 3D relative kinematics. Explain how the Earth's rotation (a rotating reference frame) induces a fictitious "Coriolis acceleration" on objects moving relative to its surface, such as large-scale air masses (wind), and describe its primary consequence on global weather patterns.

In 3D kinematics, describing the arbitrary orientation of a rigid body (like a satellite or a robotic arm's end-effector) relative to a fixed coordinate system is complex. Explain how Eulerian angles (Precession, Nutation, and Spin) provide a systematic method to specify any 3D orientation using three sequential rotations.