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

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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.