In physics, Lami's theorem is an equation relating the magnitudes of three coplanar, concurrent and non-collinear vectors, which keeps an object in static equilibrium, with the angles directly opposite to the corresponding vectors. According to the theorem,
formula_1
where formula_2 are the magnitudes of the three coplanar, concurrent and non-collinear vectors, formula_3, which keep the object in static equilibrium, and formula_4 are the angles directly opposite to the vectors, thus satisfying formula_5.
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 formula_6, hence by making all the vectors touch its tip and tail the result is a triangle with sides formula_7 and angles formula_8 (formula_4 are the exterior angles).
By the law of sines then
formula_10
Then by applying that for any angle formula_11, formula_12 (supplementary angles have the same sine), and the result is
formula_13