The parallelogram law is a part of elementary geometry. It states that “the sum of the squares of the lengths of the two diagonals of a parallelogram is same as the sum of the squares of the lengths of its four sides.”
In vector algebra, the parallelogram law is very useful. Then we commonly name it as parallelogram of vector addition.
Let P and q be two vectors. Then the sum of the vectors P + q is obtained by first placing them head to tail and then by drawing the vector from the tail that is free to the head that is free.
Then we make use of parallelogram law and say that if the vectors form the sides of the parallelogram then the vector sum and difference represents the diagonals of the parallelogram and we have
|P + q|2 + |P − q|2 = 2 |P|2 + 2 |q|2
In general, the parallelogram law of vectors states that “if the two given vectors are acting at a common point and can be represented in magnitude with direction as the two adjacent sides of a parallelogram, then the resultant vector is represented in magnitude with direction by the diagonal which is passing through the tail that is common of two vectors.”