# Orthogonality and Orthonormality

Linear algebra has a great capability for making geometric statements, due to the inherent *visual* structure in many spaces like $\mathbb{R}^n$. Line segments in such spaces are normally known as **orthogonal**, or perpendicular, if they intersect at an angle of $90^\circ$ and **orthonormal** if they additionally each have a length of $1$. In the same way, vectors are known as orthogonal if they have a dot product (or, more generally, an inner product) of $0$ and orthonormal if they have a norm of $1$. It turns out these two definitions are the same, and the connection between linear algebra and geometry quite strong.

Orthogonality (and orthonormality) is necessary to project vectors onto subspaces, find better estimates of nonlinear objects, and measure many properties of vector spaces.

#### Contents

## Normalization

The length of a vector $v$ is $\sqrt{\langle v, \, v \rangle} = \sqrt{v \cdot v}$.

## Orthogonality

Two vectors $v$ and $w$ are said to be

orthogonalif $v \cdot w = 0$. They areorthonormalif each vector has length $1$.

## Applications

Orthogonality and orthonormality allow for a much stronger geometric understanding of direction in vector spaces.

## See Also

**Cite as:**Orthogonality and Orthonormality.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/orthogonality-and-orthonormality/