Suspension Bridge Guy-Wire Force Decomposition (Civil & Mechanical Engineering)

A steel suspension bridge anchor pylon is subjected to high tensile loads from a primary stay-cable anchored to a concrete footing at $O(0, 0, 0)$ and connected to the pylon tower apex at point $P(6, 4, 12)$ (coordinates measured in meters). The cable tension exerts a force vector:

A secondary stabilizing strut extends along the direction vector $\vec{u} = \langle 2, 1, 2 \rangle$ connecting the footing to an auxiliary bedrock anchor.

To evaluate structural stability and foundation shear:

1. Decompose the cable force $\vec{F}$ into a vector component parallel to the strut ($\vec{F}_{\parallel} = \text{proj}_{\vec{u}}\vec{F}$) and an orthogonal shear force component ($\vec{F}_{\perp} = \text{orth}_{\vec{u}}\vec{F}$).
2. Determine the scalar magnitude of the compressive load along the strut and the lateral shear load.
3. Calculate the moment vector (torque) $\vec{\tau} = \vec{r} \times \vec{F}$ exerted by the cable tension about the footing origin $O$, where $\vec{r} = \vec{OP} = \langle 6, 4, 12 \rangle\text{ m}$.