 # X Coordinate|Definition & Meaning

##  Definition

On a Cartesian graph, the x-coordinate (or the abscissa) of a point represents how far along it is on the x-axis. The x-axis normally points to the left and right, so the x-coordinate tells us about the horizontal position of the point. For coordinates in the ordered pair form (x, y), the x-coordinate is always the first coordinate.

The x-coordinate 4 in (4, 3) is the signed distance from the origin to the point along the x-axis, as shown in figure 1. Figure 1 – Demonstration of the x-coordinate of a Point in an x-y-plane

## Rectangular Coordinates

The rectangular coordinates, also known as the Cartesian coordinates, are the x and y coordinates on a graph in a two-dimensional plane. They are the signed distances from the origin to the point in the horizontal and vertical directions.

The signs with the x and y coordinate specify the direction of the distance from the origin. If a point’s coordinates are (a, b), then their distance from the origin is the absolute value of a and b.

Both the x and y coordinates form the Cartesian or orthogonal coordinate system. Without the y-coordinate, the x-coordinate alone cannot trace a point in a 2D plane.

These numerical coordinates specify the position of a point by using two coordinate axes, the x and y-axis. For point A in a 2D plane, two lines, which are perpendicular to both axes, passing through point A, meet each axis at a number. These two numbers are the rectangular coordinates of point A.

The x and y coordinates are called rectangular coordinates, as locating the point through the coordinates forms a rectangle in the x-y plane. For example, point (7, 4) is located in the x-y plane, as shown in figure 2. Figure 2 – Demonstration of a Rectangle Formed by Tracing a Point by Using Rectangular Coordinates

## Ordered Pair

The rectangular coordinates are written in parentheses, separated by a comma as (x, y); the first coordinate is always the x-coordinate, and the second, is the y-coordinate. This notation (x, y) is known as an ordered pair. The x-coordinate is known as the abscissa, and the y-coordinate as ordinate.

For example, (2, 3) is an ordered pair showing the horizontal distance as 2(x-coordinate) units and the vertical distance as 3(y-coordinate) units from the origin.

Each point in the Cartesian coordinate system is specified by a unique ordered pair.

## X-Axis and Y-axis

An axis is a reference line used to measure the distance from a certain reference point. The two axes in a two-dimensional plane are the x and y-axis. The two axes are placed in such a way that they are perpendicular to each other.

The horizontal axis is known as the x-axis, while the vertical axis is the y-axis. The left-right direction is the horizontal direction, and the up-down direction is the vertical direction.

### Origin

The x and y axes can be considered as two number lines placed at right angles. The point at which both axes intersect is known as the origin. It is denoted by O with the ordered pair as (0, 0). It is the reference point from where any point can be traced by moving some units left, right, up, or down.

## Directions of Positive and Negative X and Y Axes

The direction of the positive and negative x and y axes are discussed below.

### Positive X-axis

The positive x-axis moves from the origin O from left to right. The x-coordinate increases as we move further from left to right.

For example, point (5, 3) has the x-coordinate as positive five. It means that from the origin, going five units from left to right tells how much the point is along the positive x-axis. But to completely trace the point in a 2D plane, the y-coordinate 3 should also be considered.

### Negative X-axis

The negative x-axis has a direction opposite to the positive x-axis. From origin O, the negative x-axis increases as we move from right to left. For example, point (-3, 4) has the x-coordinate as negative three. It means that from the origin, going three units from right to left shows the point along the negative x-axis.

### Positive Y-axis

The positive y-axis increases in the upper direction from the origin O. This means that the y-coordinate increases as going more and more in the upper direction. For example, point (1, 6) has a positive six as a y-coordinate. It means that going six units up from the origin shows how much the point is far up on the positive y-axis.

### Negative Y-axis

The direction of the negative y-axis is the opposite of the positive y-axis. The negative y-axis increases from the origin point in the downward direction. For example, point (2, -4) has a negative four as the y-coordinate. By moving four units down from the origin, the y-coordinate -4 is located.

Figure 3 shows the origin and the positive and negative x and y-axis. Figure 3 – Demonstration of Origin and Positive and Negative x and y-axis

Quad” means four. The x and y axes divide the Cartesian plane into four parts, known as the quadrants. The four quadrants are numbered in ascending order in a counter-clockwise direction. The four quadrants are discussed below.

In the first quadrant, both the x and y coordinates are positive. This is the top-right quadrant which includes the positive x and y-axis. For example, the ordered pair (1, 3) belongs in the first quadrant as both the x-coordinate 1 and the y-coordinate 3 are positive.

In the second quadrant, the x-coordinate is negative, and the y-coordinate is positive. This is the top-left part of the Cartesian plane, including the negative x-axis and the positive y-axis. For example, point (-1, 3) lies in the second quadrant, moving left one unit and moving up three units from the origin.

In the third quadrant, both the x and y coordinates are negative. This is the bottom-left quadrant in the x-y plane with both the x and y axis negative. For example, the ordered pair (-1, -3) belongs in the third quadrant, moving one unit left and three units down from the origin.

In the fourth quadrant, the x-coordinate is positive, and the y-coordinate is negative. This is the bottom-right quadrant with a positive x-axis and a negative y-axis. For example, point (1, -3) belongs in the fourth quadrant, moving one unit right and three units down from the origin O.

Figure 4 shows the points (1, 3), (-1, 3), (-1, -3), and (1, -3) in the first, second, third, and fourth quadrants, respectively. Figure 4 – Demonstration of the Four Quadrants and Locating a Point in all the Four Quadrants

## X-coordinate in Different Dimensions

Different dimensions require a different number of coordinates to define a point or object in space. The x-coordinate in a different number of dimensions is discussed below.

### One Dimension

One axis can be considered as a number line. A line is a one-dimensional figure, so it requires only one number to specify a point on a number line. The number line contains two half-lines, one positive half-line from to positive infinity and one negative half-line from zero to negative infinity.

### Two Dimensions

Two dimensions consist of two axes, the x and y-axis. The x and y-coordinate specify a point in a 2D plane. Two numbers are required to identify a point in a 2D plane which determines the horizontal and vertical distance from the origin O.

### Three Dimensions

A three-dimensional space consists of an x, y, and z coordinate. Three lines drawn at right angles to each other are the x, y, and z axis in space. The point where the three axes intersect is the origin O.

The x-coordinate shows how much to move left or right from the origin. The y-coordinate shows how much to move up and down from the origin, and the z-coordinate shows how much to move backward or forward to get from the origin to the point in space.

## Distance Between Two Points in an X-Y Plane

The distance between two points in an x-y plane can be found by using the x and y-coordinates of the points. If the coordinates of point C are (x1, y1) and those of point D are (x2, y2), the distance between point C and D can be calculated by using the formula:

$\text{Distance} = \sqrt{ {( y_2 \ – \ y_1 )}^2 + {( x_2 \ – \ x_1 )}^2 }$

## Examples Involving the X Coordinate

### Example 1 – Identifying X and Y Coordinates of an Ordered Pair

Point Q is shown in an x-y plane in figure 5. Identify the x and y coordinates and write the ordered pair of the point Q. Figure 5 – Identifying the x and y-coordinates of a Point in a x-y Plane

### Solution

At first, two lines that pass through point Q and are perpendicular to the two axes are drawn such that they meet the x and y axis at the x and y coordinates of point Q, respectively. Hence, the x-coordinate is 4, and the y-coordinate is 2. It can be written as an ordered pair as (4, 2).

### Example 2 – Identifying the Quadrants of Different Ordered Pairs

In which quadrants do the ordered pairs (-9, -3), (-1.4, 12), (3, 7), and (8, -5) lie?

### Solution

The point (-9, -3) lies in the third quadrant as both the x and y-coordinates are negative.

The point (-1.4, 12) lies in the second quadrant as the abscissa is negative, but the ordinate is positive.

Point (3, 7) lies in the first quadrant as both the x and y coordinates are positive.

The ordered pair (8, -5) lies in the fourth quadrant as the abscissa is positive, but the ordinate is negative.

### Example 3 – Finding the Distance Between Two Points

Find the distance between two points, T with coordinates (2, 6) and R with coordinates (5, -8).

### Solution

The formula for the distance between two points is given:

$\text{Distance} = \sqrt{ {( y_2 \ – \ y_1 )}^2 + {( x_2 \ – \ x_1 )}^2 }$

Taking the coordinates of T as (x1, y1) and the coordinates of R as (x2, y2).

So:

x1 = 2, y1 = 6, x2 = 5, y2 = -8

Putting the values in the distance formula gives:

$\text{Distance} = \sqrt{ {( – 8 \ – \ 6 )}^2 + {( 5 \ – \ 2 )}^2 }$

$\text{Distance} = \sqrt{ {( -14 )}^2 + {( 3 )}^2 }$

$\text{Distance} = \sqrt{ 196 + 9 }$

$\text{Distance} = \sqrt{ 205 }$

Distance = 14.3

All the images are created using GeoGebra.