Lami's theorem

In physics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear force vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem,

:<math>\frac{v_A}{\sin \alpha}=\frac{v_B}{\sin \beta}=\frac{v_C}{\sin \gamma}</math>

where <math>v_A, v_B, v_C</math> are the magnitudes of the three coplanar, concurrent and non-collinear vectors, <math>\vec{v}_A, \vec{v}_B, \vec{v}_C</math>, which keep the object in static equilibrium, and <math>\alpha,\beta,\gamma</math> are the angles directly opposite to the vectors, thus satisfying <math>\alpha+\beta+\gamma=360^o</math>.

Lami's theorem is applied in static analysis of mechanical and structural systems. The theorem is named after Bernard Lamy.

Proof

As the vectors must balance <math>\vec{v}_A+\vec{v}_B+\vec{v}_C=\vec{0}</math>, hence by making all the vectors touch its tip and tail the result is a triangle with sides <math>v_A,v_B,v_C</math> and angles <math>180^o -\alpha, 180^o -\beta, 180^o -\gamma</math> (<math>\alpha,\beta,\gamma</math> are the exterior angles).

By the law of sines then