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In 3D, particle equilibrium requires ΣF = 0, yielding three scalar equations: ΣFx = 0, ΣFy = 0, ΣFz = 0. Forces in 3D are expressed using Cartesian unit vectors, and cables or rod members with known geometry have their force directions determined using unit position vectors: T = T·(r_AB / |r_AB|). Setting up these unit vectors systematically from coordinate geometry is the primary skill required.
Practice writing 3D forces in Cartesian form using direction cosines or position vectors from geometry. Organize force components in a table before summing in each direction.
You already know how to solve 2D particle equilibrium — you set ΣFx = 0 and ΣFy = 0, then solve for unknowns. The 3D case adds one more equation: ΣFz = 0. In principle, this is a straightforward extension; in practice, the challenge is almost entirely geometric. Writing a cable or rod force in Cartesian component form when it points in an arbitrary direction in 3D space is where most errors occur.
The systematic approach is to find a unit position vector from the particle to the point where the cable or rod is anchored. If a cable runs from point A to point B, the position vector is r_AB = (B_x − A_x)i + (B_y − A_y)j + (B_z − A_z)k. The unit vector along that direction is û = r_AB / |r_AB|, where |r_AB| = √(Δx² + Δy² + Δz²). Then the cable force is T = T·û, giving you the three components Tx, Ty, Tz directly. This process — compute the position vector, find its magnitude, divide to get the unit vector, multiply by the force magnitude — should become automatic.
Once all forces in the problem are expressed in Cartesian form, equilibrium is mechanical: collect all the x-components and set their sum to zero, do the same for y and z. You get a system of three equations in however many unknowns you have. For a particle held by three cables, you typically have three unknown tensions — one equation per unknown. The geometry you computed at the start does all the structural work; the algebra at the end is just solving a 3×3 linear system.
A useful mental check: if you collapse the geometry to 2D (all forces in the x-y plane), your z-equation becomes 0 = 0 trivially, and the x and y equations should reproduce exactly what you would have gotten using your 2D equilibrium method. If they don't, you've made an error in the 3D setup. This check costs nothing and catches sign errors before you submit a wrong answer. The 3D skill is foundational for the space truss problems coming next, where you'll apply this exact process at every joint in a three-dimensional framework.