How can you use the parallelogram law backwards?

Answer 1

The parallelogram law states that for any two vectors ( \mathbf{u} ) and ( \mathbf{v} ), the relationship ( |\mathbf{u} + \mathbf{v}|^2 + |\mathbf{u} - \mathbf{v}|^2 = 2|\mathbf{u}|^2 + 2|\mathbf{v}|^2 ) holds true. To use this law backwards, you can take the lengths of the vectors and the resultant vector created by their sum and difference to derive the lengths of the original vectors. By rearranging the equation, you can solve for ( |\mathbf{u}| ) and ( |\mathbf{v}| ) if you know the magnitudes of ( |\mathbf{u} + \mathbf{v}| ) and ( |\mathbf{u} - \mathbf{v}| ). This approach is useful in contexts where the individual vectors are unknown but their combined effects are measurable.