Mathematics

Two vectors are perpendicular when they meet at a right angle — and the dot product captures exactly this geometric relationship. The key insight is that the dot product formula $\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta$ depends on $\cos\theta$, which equals zero precisely when $\theta = 90°$. So perpendicularity has a clean algebraic signature.

Two vectors are perpendicular when they meet at a right angle — and the dot product captures exactly this geometric relationship. The key insight is that the dot product formula $\mathbf{u} \cdot \mathbf{v} = |\mathbf{u}||\mathbf{v}|\cos\theta$ depends on $\cos\theta$, which equals zero precisely when $\theta = 90°$. So perpendicularity has a clean algebraic signature.