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Find resultant vector of the summation of two vectors A and B having D between them.

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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.

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