Industrial Valve Wrench Torque

In industrial chemical processing and piping facilities, technicians must operate heavy manual valves. Mechanical and plant engineers calculate the torque generated by hand tools to ensure operators apply adequate loosening torque without shearing valve stems or exceeding safe human mechanical leverage limits.

A technician places a wrench onto a frozen high-pressure pipe valve bolt centered at the origin $(0, 0, 0)$. The technician grips the wrench handle at a position given by $\vec{r} = \langle 0.25, 0.10, 0 \rangle\text{ meters}$ and applies a downward muscular force vector $\vec{F} = \langle 0, -45, 0 \rangle\text{ Newtons}$.

1. Compute the torque vector $\vec{\tau} = \vec{r} \times \vec{F}$ acting on the valve bolt assembly using a $3 \times 3$ determinant.
2. Determine the magnitude of the applied torque in Newton-meters ($\text{N}\cdot\text{m}$).
3. Identify the axis of rotation and describe whether the torque tends to turn the valve clockwise or counterclockwise about the $z$-axis when viewed from above.