State and Prove Parallelogram law of Vector Addition

Answer: The parallelogram law of vector addition states that, when two vectors are acting simultaneously, both in magnitude and direction, by the adjacent sides of a parallelogram drawn from the same point, then their resultant vector is represented both in magnitude and direction by the diagonal of the parallelogram also drawn from the same point.

The proof is illustrated by the following example:

Let OACB be the parallelogram.

Let P & Q be the two vectors in the adjacent sides OA, OB respectively of the parallelogram.

Let R be the resultant vector passing through the diagonal OC.

Construct a perpendicular CD to the extended OA.

Now, in the right triangle OCD, OC2=OD2+DC2

Considering the right triangle OCD, R2=(OA+AD)2+DC2

Thus, R2=OA2+AD2+2 OA.AD+DC2

From the triangle ADC, AC2=AD2+DC2

And also,  Cos θ=AD . AC

Whence, R2=OA2+AC2+2 OA.AC Cos θ

Substituting the values, we get, R2=A2+B2+2 A.B Cos θ