JEE Main & Advanced Physics Vectors Parallelogram Law of Vector Addition

If two non zero vectors are represented by the two adjacent sides of a parallelogram then the resultant is given by the diagonal of the parallelogram passing through the point of intersection of the two vectors.

(1) Magnitude 

Since, \[{{R}^{2}}=O{{N}^{2}}+C{{N}^{2}}\]

\[\Rightarrow \]\[{{R}^{2}}={{(OA+AN)}^{2}}+C{{N}^{2}}\]

\[\Rightarrow \]\[{{R}^{2}}={{A}^{2}}+{{B}^{2}}+2AB\cos \theta \] \[\therefore \]

\[R=\,|\overrightarrow{R}|\,=\,|\overrightarrow{A}+\overrightarrow{B}|\,=\sqrt{{{A}^{2}}+{{B}^{2}}+2AB\cos \theta }\]  

Special cases : \[R=A+B\] when q = 0o

\[R=A-B\] when q = 180o

\[R=\sqrt{{{A}^{2}}+{{B}^{2}}}\] when q = 90o

(2) Direction

\[\tan \beta =\frac{CN}{ON}=\frac{B\sin \theta }{A+B\cos \theta }\]