2.9a Orthogonal Vectors

Orthogonality

A set of vectors $S\in {\mathbb{R}}^n$ is pairwise orthogonal if and only if every distinct pair of vectors $\vec{x},\vec{y}\in S$ is pairwise orthogonal, such that
$⟨\vec{x},\vec{y}⟩=0.$
If $S$ is orthogonal, then it is linearly independent and contains at most $n$ vectors.

Two sets of vectors $S_1$ and $S_2$ are mutually orthogonal if:

• All $\vec{x}\in S_1$ and $\vec{y}\in S_2$ are pairwise orthogonal.

• The bases $B_1$ and $B_2$ of $S_1$ and $S_2$ are orthogonal to each other.

Orthogonality in Euclidean Space $\vec{x}$ and $\vec{y}$ are orthogonal if they are perpendicular in Euclidean space, such that they make an angle of $\theta =90°$ between them. This is critical in order to meet the dot product criterion $|\vec{x}||\vec{y}|\cos \theta =0$.

Two nonzero vectors