# State parallelogram law of vector addition and derive an expression for its magnitude.

Shoaib Malik
20 Points
7 years ago
Parallelogram Law of Vector AdditionStatement of Parallelogram Law If two vectors acting simultaneously at a point can be represented both in magnitude and direction by the adjacent sides of a parallelogram drawn from a point, then the resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram passing through that point.Derivation of the lawNote: All the letters in bold represent vectors and normal letters represent magnitude only.Let P and Q be two vectors acting simultaneously at a point and represented both in magnitude and direction by two adjacent sides OA and OD of a parallelogram OABD as shown in figure.Let θ be the angle between P and Q and R be the resultant vector. Then, according to parallelogram law of vector addition, diagonal OB represents the resultant of P and Q.So, we have R = P + QNow, expand A to C and draw BC perpendicular to OC.From triangle OCB, In triangle ABC, Also, Magnitude of resultant:Substituting value of AC and BC in (i), we get which is the magnitude of resultant.Direction of resultant: Let ø be the angle made by resultant R with P. Then,From triangle OBC, which is the direction of resultant
Shoaib Malik
20 Points
7 years ago