Lami’s theorem in vector:
Lami’s theorem is used to relate forces with angle between them when three coplanar forces are acting from a point in equilibrium. The ratio of one force to the sine of angle between other two forces is constant.
Statement and prove:
According to Lami’s theorem that for a triangle, the ratio of one side with the sign of opposite angle is constant. For a triangle 
ABC, 
= 
= 
.
Let us consider, AD is perpendicular to BC.
From ABD, 
= sinB and from ACD, 
= sinC so, ABsinB = ACsinC or, = —– (i)
Let CE is perpendicular to AB.
From AEC, 
= sinA and from BCE, 
= sinB so, ACsinA = BCsinB or, = —– (ii)
From equation (i) and (ii) we get, = = .
Example of Lami’s theorem:
1. A bob of mass 2kg is suspended freely from the celling by a string. Now a horizontal force F is applied on the bob and the string creates angle 
with vertical. What is the value of tension acts on the string and F if the bob is at rest?
The angle between tension T and F is 
, the angle between F and weight of the bob (Mg = 20N) is 
and the angle between Mg and T is 
. Using Lami’s theorem we get, 
= 
= 
.
Now =
Or, 
= 
F = 20tan = 
.
Again = or, T = = 
.
- The resultant of two vectors P and
is inclined at an angle θ with . Show that the value of θ is not greater than 
.
Using Lami’s theorem for ∆OCA, 
= 
or, 
= 
or, sin
= 
We know that, the value of sin
≤ 1
When sin = 1 then, sin = 
and if sin < 1 then, sin <
Therefore, sin ≤ or, sin ≤ sin or, θ ≤ .
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