The energy of a system can be expressed as the sum of the energies of its normal modes due to the orthogonality of mode shapes. When a system is excited, its motion can be represented as a linear combination of these modes, allowing for a straightforward calculation of total energy. The mathematical proof of orthogonality relies on linear algebra, demonstrating that the inner products of different modes yield zero when mass and stiffness matrices are symmetric. This principle simplifies the analysis of complex motions, such as those in vibrating systems, by enabling predictions based on fundamental and harmonic frequencies. Understanding this concept is crucial for practical engineering applications, even if the detailed proof is not always necessary.