Question

State parallelogram law of vectors. Derive an expression for the magnitude and direction of the resultant vector.

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Answer 1

Answer:

Parallelogram law states that if two vectors are considered to be the adjacent sides of a Parallelogram, then the resultant of two vectors is given by the vector which is a diagonal passing through the point of contact of two vectors.

In the figure P and Q are two vectors.with magnitudes equal to length OA and OB respectively and making angle θ between them. Complete the parallelogram, OACB,

Join diagonal OC , that makes angle α with vector P.

According to parallelogram law of vectors the resultant is represented by the diagonal  passing through the point of contact of two vectors.

To find the magnitude of resultant , produce a perpendicular CD to meet OA produced to D.

From △ OCD,

OC2=OD2+CD2

Now CD=AC sinθ=Qsinθ

AD=ACcosθ=Qcosθ

OD=OA+AD=P+Qcosθ

Putting these values and representing resultant vector OC by R, magnitude of the resultant is given by

R2=(P+Qcosθ)2+(Qsinθ)2=P2+Q2+2PQ