6.4 Orthogonality

Besides calculating lengths of vectors and angles between vectors, an inner product allows us to know whether two vectors are orthogonal. In a two dimensional space, orthogonality is equivalent to perpendicularity; so if two vectors are perpendicular to each other—the angle between them is a 90 degree angle—they are orthogonal. Two vectors vectors \(\mathbf{x}\) and \(\mathbf{y}\) are orthogonal if their inner product is zero:

\[ \mathbf{x^\mathsf{T} y} = 0 \iff \mathbf{x} \hspace{1mm} \bot \hspace{1mm} \mathbf{y} \]