Orthogonality proof demonstrating when the sum and difference of two vectors have equal magnitude in abstract vector spaces. The diagram walks through a rigorous algebraic proof starting from the hypothesis ||u+v|| = ||u-v||, squaring both sides, expanding using inner products, and simplifying to derive u·v = 0. This proof establishes the fundamental geometric principle that two vectors are orthogonal if and only if the diagonals of their parallelogram are equal in length. Fork this diagram to customize the proof steps, add geometric visualizations, or adapt it for teaching linear algebra and functional analysis courses.