Orthogonal Vectors in an Inner Product Space

Definition: Let $H$ be an inner product space. Two elements $x, y \in H$ are said to be Orthogonal or Perpendicular denoted $x \perp y$ if $\langle x, y \rangle = 0$.

For example, the vectors $(1, 2, 3)$ and $(2, -1, 0)$ are orthogonal vectors in $\mathbb{R}^3$ since:

(1)

\begin{align} \quad \langle (1, 2, 3), (2, -1, 0) \rangle = (1)(2) + (2)(-1) + (3)(0) = 2 - 2 + 0 = 0 \end{align}

Definition: Let $H$ be an inner product space, $x \in H$, and $S \subseteq H$. Then $x$ is said to be Orthogonal to $S$ denoted $x \perp S$ if $\langle x, y \rangle = 0$ for every $y \in S$.

For example, consider the space $\mathbb{R}^2$ and the vector $(1, 1) \in \mathbb{R}^2$. Let $(x, y) \in \mathbb{R}^2$ and consider the inner product:

(2)

\begin{align} \quad \langle (1, 1), (x, y) \rangle = 1 \cdot x + 1 \cdot y = x + y \end{align}

Then $(1, -1)$ orthogonal to $(1, 1)$, that is:

(3)

\begin{align} \quad (1, 1) \perp (1, -1) \end{align}

Furthermore, if $S = \{ (x, y) \in \mathbb{R}^2 : x + y = 0 \}$, then:

(4)

\begin{align} \quad (1, 1) \perp S \end{align}