Orthogonality

In $\mathbb{R}^k$ with the standard inner product, two vectors $x,y\in\mathbb{R}^k$ are orthogonal, written $x\perp y$, if

Orthogonality generalizes the geometric notion of “perpendicular.” It is a key concept in decompositions (e.g., Pythagorean theorem, orthonormal bases) and in analysis via projections and least squares.

Examples:

• In $\mathbb{R}^2$, $(1,0)\perp(0,1)$.
• In $\mathbb{R}^3$, $(1,1,0)\perp(1,-1,0)$ since $1\cdot 1 + 1\cdot(-1)+0\cdot 0=0$.
• A nonzero vector $x$ is never orthogonal to itself, since $\langle x,x\rangle>0$.