Editing Resultant force (section)

==Associated torque==
If a point '''R''' is selected as the point of application of the resultant force '''F''' of a system of ''n'' forces '''F'''<sub>i</sub> then the associated torque '''T''' is determined from the formulas
:<math> \mathbf{F} = \sum_{i=1}^n \mathbf{F}_i,</math>
and
:<math> \mathbf{T} = \sum_{i=1}^n (\mathbf{R}_i-\mathbf{R})\times \mathbf{F}_i. </math>

It is useful to note that the point of application '''R''' of the resultant force may be anywhere along the [[line of action]] of '''F''' without changing the value of the associated torque.  To see this add the vector k'''F''' to the point of application '''R''' in the calculation of the associated torque,
:<math> \mathbf{T} = \sum_{i=1}^n (\mathbf{R}_i-(\mathbf{R}+k\mathbf{F}))\times \mathbf{F}_i. </math>
The right side of this equation can be separated into the original formula for '''T''' plus the additional term including k'''F''',
:<math> \mathbf{T} = \sum_{i=1}^n (\mathbf{R}_i-\mathbf{R})\times \mathbf{F}_i - \sum_{i=1}^n k\mathbf{F}\times \mathbf{F}_i=\sum_{i=1}^n (\mathbf{R}_i-\mathbf{R})\times \mathbf{F}_i,</math>
because the second term is zero.  To see this notice that '''F''' is the sum of the vectors '''F'''<sub>i</sub> which yields
:<math>\sum_{i=1}^n k\mathbf{F}\times \mathbf{F}_i = k\mathbf{F}\times(\sum_{i=1}^n \mathbf{F}_i )=0,</math>
thus the value of the associated torque is unchanged.