Robotic Arm Joint Torque

In industrial automation and robotics, multi-axis robotic arms manipulate heavy tools and end-effectors in 3D manufacturing space. Robotics and computer systems engineers calculate cross-product torque vectors at each articulated joint to drive servomotor controllers and prevent structural overload.

A 6-axis industrial robot has its base shoulder joint pivot located at the Cartesian origin $(0, 0, 0)$. When holding an end-effector welding tool, the position vector from the shoulder pivot to the tool's center of mass is $\vec{r} = \langle 0.8, 0, 0.6 \rangle\text{ meters}$. The gravitational downward weight of the tool exerts a constant vertical force vector $\vec{F} = \langle 0, -100, 0 \rangle\text{ Newtons}$.

1. Compute the structural torque vector $\vec{\tau} = \vec{r} \times \vec{F}$ experienced at the shoulder pivot joint using a matrix determinant.
2. Calculate the magnitude of the torque ($|\vec{\tau}|$) in Newton-meters ($\text{N}\cdot\text{m}$).
3. Identify which Cartesian axes experience bending/torsion moments.