X-Coordinate
Geometry & MeasurementThe x-coordinate is the first number in an ordered pair, indicating a point's horizontal distance and direction from the origin.
Definition
The x-coordinate is the first number in an ordered pair $(x, y)$. It tells you how far to move left or right from the origin. Positive means right; negative means left.
Example
In the point $(6, 2)$, the x-coordinate is $6$, meaning you move $6$ spaces to the right. In the point $(-4, 3)$, the x-coordinate is $-4$, meaning you move $4$ spaces to the left.
Key Insight
The x-coordinate is sometimes called the "abscissa." Always read an ordered pair left to right: x first, y second. Think "x comes before y in the alphabet" to remember the order.
Definition
The x-coordinate (abscissa) of a point $(x, y)$ is its horizontal displacement from the origin: positive values lie to the right of the y-axis, negative values to the left. For a function $f$, the x-coordinate represents the input (independent variable).
Example
To find the x-coordinate of the midpoint of $(2, 8)$ and $(10, 4)$: $x_{mid} = (2 + 10)/2 = 6$. The midpoint is $(6, 6)$. The x-coordinate of the midpoint is the average of the two x-coordinates.
Key Insight
The x-coordinate is the projection of a point onto the x-axis. Geometrically, drop a perpendicular from the point to the x-axis; where it lands is the x-coordinate. This projection idea extends to higher-dimensional spaces.
Definition
The x-coordinate of a point $p$ in $\mathbb{R}^2$ is the result of applying the linear projection $\pi_1 : \mathbb{R}^2 \to \mathbb{R}$ defined by $\pi_1(x, y) = x$. This is the dot product of $p$ with the unit vector $e_1 = (1, 0)$. In a rotated coordinate system with basis $\{u, v\}$, the "x-coordinate" becomes $p \cdot u / |u|$.
Example
Under a $45$-degree rotation, the new x-coordinate of point $(3, 1)$ in the rotated frame is $3\cos(45^\circ) + 1\sin(45^\circ) = (3 + 1)/\sqrt{2} = 4/\sqrt{2} = 2\sqrt{2}$.
Key Insight
Coordinates are not intrinsic properties of points but depend on the choice of basis. Changing basis (e.g., rotating axes) changes all coordinates while the geometric point remains fixed. This is the core insight of the distinction between vectors and coordinates.