Equilibriant Vector
A single vector which balances two or more vectors acting on a body at the same time is called an equilibrant vector. It is equal and opposite to the resultant of various vectors acting on the body at the same time.
Lamis theorem (application of triangle law)
"If three vectors acting at a point are in equilibrium, then magnitude of each vector is proportional to the sine of the angle between the other two vectors"
i.e., \(
\frac{A}
{{\sin \alpha }} = \frac{B}
{{\sin \beta }} = \frac{C}
{{\sin \gamma }}
\)
where \(
\vec A,\vec B,\vec C
\) are the three forces and \(
\alpha ,\beta ,\gamma
\) are the angle between forces \(\vec B\) and \(\vec C\), \(\vec C\)and \(\vec A\) and \(\vec A\) and respectively, as in below figure
Proof : Let three forces , and acting at a point be in equilibrium. They must be represented by the three sides of a triangle OPQ, taken in an order.
Therefore, \(
\frac{A}
{{OP}} = \frac{B}
{{PQ}} = \frac{C}
{{QO}}
\)
From the triangle OPQ, by sine formula, we have \(
\frac{{OP}}
{{\sin \angle PQO}} = \frac{{PQ}}
{{\sin \angle QOP}} = \frac{{QO}}
{{\sin \angle OPQ}}
\)
Then we have \(
\frac{A}
{{\sin \angle PQO}}
\) = \(
\frac{B}
{{\sin \angle QOP}}
\) = \(
\frac{C}
{{\sin \angle OPQ}}
\)
(or) \(
\frac{A}
{{\sin (180^\circ - \alpha )}}
\) =\(
\frac{B}
{{\sin (180^\circ - \beta )}}
\) = \(
\frac{C}
{{\sin (180^\circ - \gamma )}}
\)
(or) \(
\frac{A}
{{\sin \alpha }}
\) = \(
\frac{B}
{{\sin \beta }}
\) = \(
\frac{C}
{{\sin \gamma }}
\)