Once you have formulated
\(\Sigma\vec{F}=0\) and
\(\Sigma\vec{M}=0\) equations in each of the
\(x\text{,}\) \(y\) and
\(z\) directions, you could be facing up to six equations and six unknown values.
Frequently these simultaneous equation sets can be solved with substitution, but it is often be easier to solve large equation sets with linear algebra. Note that the adjective “linear” specifies that the unknown values must be linear terms, which means that each unknown variable cannot be raised to a exponent, be an exponent, or buried inside of a
\(\sin\) or
\(\cos\) function. Luckily, most unknowns in equilibrium are linear terms, except for unknown angles. If you are not familiar with the use of linear algebra matrices to solve simultaneously equations, search the internet for
Solving Systems of Equations Using Linear Algebra and you will find plenty of resources.
No matter how you choose to solve for the unknown values, any numeric values which come out to be negative indicate that your initial hypothesis of that vector’s sense was incorrect.