Orthogonal Vectors¶

Definition¶

Two vectors $\boldsymbol{x}$ and $\boldsymbol{y}$ are orthogonal if their inner product is zero: $\boldsymbol{x}\cdot \boldsymbol{y}=0$ $0$ .

Interpretation¶

Two nonzero orthogonal vectors meet at a right angle in Euclidean geometry. The inner product turns this geometric condition into a computation: instead of measuring an angle, multiply matching coordinates, add, and check whether the result is $0$ .

The vectors x = (3, 1) and y = (-1, 3) on a grid, meeting at a right angle at the origin, since their dot product is 0.

The vectors $\boldsymbol{x}=(3,1)$ and $\boldsymbol{y}=(-1,3)$ in the figure satisfy $\boldsymbol{x}\cdot\boldsymbol{y}=3\cdot(-1)+1\cdot3=0$ $0$ , so the angle between them is $90^\circ$ . Here $\boldsymbol{y}$ is $\boldsymbol{x}$ turned a quarter turn; in $\mathbb{R}^2$ the vectors $(a,b)$ and $(-b,a)$ are always orthogonal. Pythagoras holds as well: $\boldsymbol{x}+\boldsymbol{y}=(2,4)$ has $\|\boldsymbol{x}+\boldsymbol{y}\|^2=20=10+10=\|\boldsymbol{x}\|^2+\|\boldsymbol{y}\|^2$ $∥ y ∥^{2}$ .

Turn $\boldsymbol{y}$ until $\boldsymbol{x}\cdot \boldsymbol{y}=0$ $0$ , or press the button: for two nonzero vectors, the shadow of $\boldsymbol{x}$ on the line of $\boldsymbol{y}$ disappears and they meet at a right angle. A zero vector is also orthogonal by definition, but has no angle.

The link with Pythagoras holds in every dimension. Expanding gives $\|\boldsymbol{x}+\boldsymbol{y}\|^2=\|\boldsymbol{x}\|^2+2\,\boldsymbol{x}\cdot\boldsymbol{y}+\|\boldsymbol{y}\|^2$ $∥ y ∥^{2}$ , so the Euclidean norms add in squares exactly when $\boldsymbol{x}\cdot\boldsymbol{y}=0$ $0$ . The definition also covers the zero vector: it is orthogonal to every vector, and it is the only vector orthogonal to itself, because $\boldsymbol{x}\cdot\boldsymbol{x}=\|\boldsymbol{x}\|^2$ $∥ x ∥^{2}$ . A right angle only makes sense for nonzero vectors.

Orthogonality defines the normal vector of a plane, and the shortest distance from a point to a line is measured along the perpendicular. It returns later in orthonormal bases, bases of mutually orthogonal vectors of length $1$ .

Orthogonal should not be confused with linearly independent. Nonzero vectors that are pairwise orthogonal are always independent, but independent vectors need not be orthogonal: $(1,0)$ and $(1,1)$ are independent, with inner product $1$ .

Key Results¶

Orthogonality is equivalent to a $90^\circ$ angle for nonzero real vectors.
If $\boldsymbol{y}$ has unit length, then $\boldsymbol{x}\cdot \boldsymbol{y}$ gives the signed length of the projection of $\boldsymbol{x}$ onto $\boldsymbol{y}$ .

Examples¶

$(1,0)$ and $(0,1)$ are orthogonal.
$(1,1)$ and $(1,-1)$ are orthogonal.

Prerequisites¶

Inner Product

Sources¶

Math II: Linear Algebra · Week 5

Aliases: perpendicular vectors Keywords: orthogonality Symbols: real numbers

Last updated: 27 October 2026 11:52