Orthogonal vectors

Definition · applies to vectors in an inner product space · also called orthogonal vectors

High school

Two vectors in the plane or in space are orthogonal if the vectors meet at a right angle, which happens exactly when their dot product is $0$.

Undergraduate (inner product spaces)

Consider an inner product space $V$. Two vectors $u, v \in V$ are orthogonal if

\[\ip{u}{v} = 0.\]

A list of vectors is orthogonal if every two different vectors in the list are orthogonal.

Examples

The lines $y = x$ and $y = -x$ in the plane, with direction vectors $(1, 1)$ and $(1, -1)$ High school
The direction vectors $(1, 1)$ and $(1, -1)$ of the lines $y = x$ and $y = -x$ are orthogonal, because their dot product is $1 - 1 = 0$.
The $x$-axis and the line through $(0, 0, 1)$ parallel to the $y$-axis in $\reals^3$ High school
The $x$-axis and the line through $(0, 0, 1)$ parallel to the $y$-axis have orthogonal direction vectors $(1, 0, 0)$ and $(0, 1, 0)$, so the lines are orthogonal.
The vectors $(1, 0)$ and $(0, 1)$ in $\reals^2$ High school
The vectors $(1, 0)$ and $(0, 1)$ are orthogonal, because their dot product is $1 \cdot 0 + 0 \cdot 1 = 0$.
The vector $(1, 1)$ and the line $x + y = 1$ in the plane, whose direction vector is $(1, -1)$ High school
The vector $(1, 1)$ is orthogonal to the direction vector $(1, -1)$ of the line $x + y = 1$, because their dot product is $1 - 1 = 0$.
The functions $\sin$ and $\cos$ on $[-\pi, \pi]$, with the $L^2$ inner product $\ip{f}{g} = \int_{-\pi}^{\pi} f(x) g(x) \, dx$ Undergraduate
The functions $\sin$ and $\cos$ are orthogonal in the $L^2$ inner product on $[-\pi, \pi]$, because $\int_{-\pi}^{\pi} \sin(x) \cos(x) \, dx = 0$, since the integrand is odd.
The vectors $(1, 0)$ and $(0, 0)$ in $\reals^2$ Undergraduate
The vectors $(1, 0)$ and $(0, 0)$ are orthogonal, because their dot product is $1 \cdot 0 + 0 \cdot 0 = 0$.

Things that may be mistaken for examples.

The vectors $(1, 0)$ and $(2, 0)$ in $\reals^2$ High school
The vectors $(1, 0)$ and $(2, 0)$ are not orthogonal, because their dot product is $1 \cdot 2 + 0 \cdot 0 = 2$.

Linearly independent vs. orthogonal: A list of vectors can point in “different directions” in two senses. A set of linearly independent vectors have no redundant vectors: no vector in the list is a linear combination of the others. Orthogonal vectors are pairwise at right angles: every two different vectors in the list have inner product $0$. Linear independence is defined in every vector space, but orthogonality requires an inner product.
Right angle vs. perpendicular vs. orthogonal vs. normal: Right angles, perpendicular lines, orthogonal vectors, and normal vectors all describe things that meet “squarely.” A right angle is an angle of $90^\circ$, so the term describes an angle. Perpendicular describes lines or planes that intersect and form right angles. Orthogonal describes vectors whose inner product is $0$, which extends the idea to vectors that do not meet, such as directions of skew lines, to the zero vector, and to vectors that are not arrows, such as functions. A normal vector is perpendicular to a line, curve, or surface at a point, so the term describes a vector relative to a shape.