Coordinate¶

Definition¶

In $\mathbb{R}^n$ , if

\boldsymbol{v}=(v_1,v_2,\ldots,v_n),

then $v_i$ is called the $i$ th coordinate of $\boldsymbol{v}$ .

Interpretation¶

The $i$ th coordinate of $\boldsymbol{v}=(v_1,\ldots,v_n)$ is the number in position $i$ of the tuple: it says how far to go along the $i$ th axis. Coordinates are the individual entries of a vector in real coordinate space, and two vectors of $\mathbb{R}^n$ are equal if and only if all their coordinates agree.

A vector w on a grid, written as 3.8 e1 + 2 e2 in the standard basis and as 2 b1 + b2 in a second basis with vectors b1 and b2.

The figure shows that coordinates depend on the chosen basis. In the standard basis $\mathcal{E}=(\boldsymbol{e}_1,\boldsymbol{e}_2)$ $(e_{1}, e_{2})$ , the dashed grey lines drop from $\boldsymbol{w}$ to $3.8$ on the $x_1$ -axis and to $2$ on the $x_2$ -axis, so $[\boldsymbol{w}]_{\mathcal{E}}=(3.8,2)$ and $\boldsymbol{w}=3.8\boldsymbol{e}_1+2\boldsymbol{e}_2$ $2 e_{2}$ . Along the dashed blue path, the same vector is $\boldsymbol{w}=2\boldsymbol{b}_1+\boldsymbol{b}_2$ $b_{2}$ , so in the basis $\mathcal{B}=(\boldsymbol{b}_1,\boldsymbol{b}_2)$ it has coordinates $[\boldsymbol{w}]_{\mathcal{B}}=(2,1)$ $(2, 1)$ .

Drag $\boldsymbol{v}$ and watch $v_1$ and $v_2$ : each coordinate is the step taken along one axis.

In a general vector space the vectors need not be tuples, and coordinates are defined through a basis: they are the coefficients of the unique linear combination of the basis vectors that gives the vector. The standard basis of $\mathbb{R}^n$ makes those coefficients exactly the entries $v_i$ , since $\boldsymbol{v}=v_1\boldsymbol{e}_1+\cdots+v_n\boldsymbol{e}_n$ $v_{n} e_{n}$ . For polynomials of degree at most $2$ with the basis $(1,x,x^2)$ , the coordinates of $3-x+4x^2$ are $(3,-1,4)$ . In the plane, the Cartesian coordinates of a point are the coordinates of the arrow from the origin to that point in the standard basis.

A coordinate is a number, not a vector: $v_1$ is a scalar, while $v_1\boldsymbol{e}_1$ is the vector that goes $v_1$ steps along the first axis.

Key Results¶

A vector in $\mathbb{R}^n$ has exactly $n$ coordinates.

Examples¶

For $\boldsymbol{v}=(3,-1,4)$ $(3, - 1, 4)$ , the second coordinate is $-1$ .

Prerequisites¶

Vector Space

Sources¶

Math II: Linear Algebra · Week 2

Aliases: coordinate tuple Keywords: component, coordinates Symbols: real numbers

Last updated: 20 October 2026 03:02