Structural Load Decomposition

In civil and structural engineering, complex loads acting on bridge trusses and framework joints must be resolved along individual steel members. By projecting external force vectors onto member axes, engineers determine whether structural beams are under tension or compression.

A structural node on a pedestrian bridge truss experiences a downward gravitational load vector of $\vec{F}_{\text{load}} = \langle 0, -50 \rangle\text{ kN}$. One of the diagonal steel support beams framing into this junction is oriented along the vector direction $\vec{a} = \langle 4, 3 \rangle$.

1. Find the unit vector $\hat{u}_a$ pointing in the direction of the support beam $\vec{a}$.
2. Compute the scalar component ($\text{comp}_{\vec{a}}\vec{F}_{\text{load}}$) of the load directed along the axis of this support beam.
3. Find the vector projection ($\text{proj}_{\vec{a}}\vec{F}_{\text{load}}$) of the load onto the support beam.