There are six equations expressing the equilibrium of a rigid body in 3 dimensions.

Sum of Forces: ${\displaystyle \sum _{}^{}F_{x}=0}$, ${\displaystyle \sum _{}^{}F_{y}=0}$, ${\displaystyle \sum _{}^{}F_{z}=0}$

Sum of Moments: ${\displaystyle \sum _{}^{}M_{x}=0}$, ${\displaystyle \sum _{}^{}M_{y}=0}$, ${\displaystyle \sum _{}^{}M_{z}=0}$

To solve three dimensional statics problems:

• Resolve force components
• Resolve moments about a point
• Resolve moments about a line or axis. See Scalar Triple Product

Knowing how to find the Determinate of a matrix can help a lot here.