# Definition:Orthogonal (Linear Algebra)

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## Definition

Let $\left({V, \left\langle {\cdot} \right\rangle}\right)$ be an inner product space.

Let $u, v \in V$.

Then $u$ and $v$ are **orthogonal** if and only if:

- $\left\langle{ u, v }\right\rangle = 0$

### Orthogonal Set

Let $S = \left\{{u_1, \ldots, u_n}\right\}$ be a subset of $V$.

Then $S$ is an **orthogonal set** iff its elements are pairwise orthogonal:

- $\forall i \ne j: \left\langle {u_i, u_j} \right\rangle = 0$

### Vectors in $\R^n$

Let $\mathbf{u}$, $\mathbf{v}$ be vectors in $\R^n$.

Then $\mathbf{u}$ and $\mathbf{v}$ are said to be **orthogonal** iff their dot product is zero:

- $\mathbf{u} \cdot \mathbf{v} = 0$

As Dot Product is Inner Product, this is a special case of the definition of orthogonal vectors.

## Also see

## Sources

- For a video presentation of the contents of this page, visit the Khan Academy.