Lami's theorem

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,


 * $$\frac{v_A}{\sin \alpha}=\frac{v_B}{\sin \beta}=\frac{v_C}{\sin \gamma}$$

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

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

By the law of sines then

$$\frac{v_A}{\sin (180^o -\alpha)}=\frac{v_B}{\sin (180^o-\beta)}=\frac{v_C}{\sin (180^o-\gamma)}.$$

Then by applying that for any angle $$\theta$$, $$\sin (180^o - \theta) = \sin \theta$$ (suplementary angles have the same sine), and the result is

$$\frac{v_A}{\sin \alpha}=\frac{v_B}{\sin \beta}=\frac{v_C}{\sin \gamma}.$$