Let there be two vectors \(\vec A\) and \(\vec B\) with D between them.
Using parallelogram law of vector addition resultant.
\(\vec R=\vec A+\vec B\)
Raw DF ⊥ OC extended as OF
Considering right angled ∆OFD
OD2 = OF2 + DF2
= (OC + CF)2 + DF2
= (A + Bcos θ)2 + (Bsin θ)2
R2 = A2 + B2 + 2ABcos θ (∵ cos2θ + sin2 θ = 1)
R = \(\sqrt{A^2+B^2+2ABcos\,\theta}\) …(i)
tan α = \(\frac{DF}{OF}=\frac{DF}{OC+CF}\)
= \(\frac{B\,sin \,\theta}{A+B\,cos\,\theta}\)
α = \(tan^{-1}\frac{B\,sin \,\theta}{A+B\,cos\,\theta}\) ...(ii)
Eqn. (i) is called law of cosines.
