Editing Resultant force (section)

==Wrench==
The forces and torques acting on a rigid body can be assembled into the pair of vectors called a [[Wrench (screw theory)|''wrench'']].<ref>R. M. Murray, Z. Li, and S. Sastry, [https://books.google.com/books?id=D_PqGKRo7oIC&dq=screw+theory+wrench&pg=PA19 ''A Mathematical Introduction to Robotic Manipulation,''] CRC Press, 1994</ref> If a system of forces and torques has a net resultant force '''F''' and a net resultant torque '''T''', then the entire system can be replaced by a force '''F''' and an arbitrarily located couple that yields a torque of '''T'''. In general, if '''F''' and '''T''' are orthogonal, it is possible to derive a radial vector '''R''' such that <math> \mathbf{R}\times\mathbf{F} = \mathbf{T} </math>, meaning that the single force '''F''', acting at displacement '''R''', can replace the system.  If the system is zero-force (torque only), it is termed a ''screw'' and is mathematically formulated as [[screw theory]].<ref>[https://books.google.com/books?id=Qu9IAAAAMAAJ&dq=The%20theory%20of%20screws%3A%20A%20study%20in%20the%20dynamics%20of%20a%20rigid%20body&pg=PR3 R. S. Ball, ''The Theory of Screws: A study in the dynamics of a rigid body'', Hodges, Foster & Co., 1876]</ref><ref>[https://books.google.com/books?id=jv9mQyjRIw4C&q=geometric+design+of+linkages  J. M. McCarthy and G. S. Soh, ''Geometric Design of Linkages''. 2nd Edition, Springer 2010]</ref>
  
The resultant force and torque on a rigid body obtained from a system of forces '''F'''<sub>i</sub> i=1,...,n, is simply the sum of the individual wrenches W<sub>i</sub>, that is

:<math> \mathsf{W} = \sum_{i=1}^n \mathsf{W}_i = \sum_{i=1}^n (\mathbf{F}_i, \mathbf{R}_i\times\mathbf{F}_i). </math>
Notice that the case of two equal but opposite forces '''F''' and '''-F''' acting at points '''A''' and '''B''' respectively, yields the resultant W=('''F'''-'''F''',  '''A'''×'''F''' - '''B'''× '''F''') = (0, ('''A'''-'''B''')×'''F''').  This shows that wrenches of the form W=(0, '''T''') can be interpreted as pure torques.