Please draw a parallelogram AB CD. Let CD be parallel to AB. Let BC be parallel to AD. Let vector P be represented by AB. Let Q be represented by AD. Now the resultant force of P & Q is given by parallelo gram law as
R = square root [ P²+ Q² + 2 P Q cos A ] = This is diagonal AC
If the angle is reversed, That is, reverse vector AD or Q. Now draw parallelogram AEFB so that AE = - Q = - AD. FB = EA. Now the diagonal AF will be parallel to diagonal BD. The angle at A in AEFB is 180 - A. SO its cosine is - Cos A.
S = squareroot [ P² + Q² - 2 P Q cos A ]
So R² + Q² = 2 [P² + Q² ]
as the other terms cancel.
This is also a relation between sides of a parallelogram and diagonals
