---
tag: 0TS7VZJBWD
kind: definition
name: Orthogonal vectors
msc: [15A63, 46C05]
defines: 0Y2X7P2WB7
uses: [0VGSYBWHV2]
sources:
  - url: https://en.wikipedia.org/wiki/Orthogonality_(mathematics)
    locator: Euclidean vector spaces
---

## Definition

Let $V$ be an inner product space over $\R$ or $\C$.
Two vectors $x, y \in V$ are *orthogonal*, written $x \perp y$, when $\langle x, y \rangle = 0$, where $\langle \cdot, \cdot \rangle$ is the [[inner product|0NB0AWGJBW]].

## Remarks

The relation is symmetric, because $\langle y, x \rangle = \overline{\langle x, y \rangle}$.
The zero vector is orthogonal to every vector, and it is the only vector orthogonal to itself.
In $\R^2$ with the dot product, $(1, 2)$ and $(-2, 1)$ are orthogonal.

## Explanations

### L1 human

Orthogonal is the general word for perpendicular.
Two vectors are orthogonal when their inner product is zero.
In the plane and in space, this means that they meet at a right angle.