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## State and prove Varigonâ€™s theorem.-Elements of Civil Engineering and Mechanics

Elements of Civil Engineering and MechanicsAsked by:milan-ransingh

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Aparna-Dasguptaprajwalamv

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AnswersFor a set

nof vectors ${\boldsymbol {u_{1}}},{\boldsymbol {u_{2}}},...,{\boldsymbol {u_{n}}}$ that concurs at pointOin space. The resultant is:${\boldsymbol {R}}=\sum _{i}{\boldsymbol {u_{i}}}$

The moment of each vector is:

$\sum _{i}{\boldsymbol {M_{O_{1}}^{u_{i}}}}=\sum _{i}({\boldsymbol {O}}-{\boldsymbol {O_{1}}})\times {\boldsymbol {u_{i}}}$

In terms of summary, taking out the common factor $(\mathbf {O} -\mathbf {O_{1}} )$, and considering the expression of

R, results:$\sum _{i}{\boldsymbol {M_{O_{1}}^{u_{i}}}}=({\boldsymbol {O}}-{\boldsymbol {O_{1}}})\times \left(\sum _{i}{\boldsymbol {u_{i}}}\right)=({\boldsymbol {O}}-{\boldsymbol {O_{1}}})\times {\boldsymbol {R}}={\boldsymbol {M_{O_{1}}^{R}}}$

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