How to Write and Read Coordinates on the X and Y Axes
We can define the Cartesian coordinate system as a coordinate system that uniquely defines each point in a plane by mean of a set of numerical coordinates, which are the distances signed from two fixed perpendicular oriented lines to the point measured in the same unit of length.
In other words, for any point on that surface, two lines drawn at right angles to each other on a flat surface provide a reference grid. Since the reference axes are perpendicular, the cartesian coordinate system is also called a rectangular or orthogonal coordinate system.
A standard Cartesian coordinate system is defined by the x-axis (Horizontal number line) and y-axis (Vertical number line). A unit of length or distance for each axis is metres or miles. A distance relative to both the x and y axes is defined by any point within the coordinate system (x,y). At the point where the value of both x and y is zero is called the origin (0,0).
Sometimes the horizontal (x) axis is called the abscissa and the vertical (y) axis is called the ordinate. The abscissa and the ordinate, regardless of whether the axes are called x and y or something else, are the first and second coordinates of every point in the coordinate system.
Writing Coordinates on X-Y Plot
Writing coordinates on the plot is enclosed between the brackets with a comma (,) separating the two coordinate values i.e horizontal (x) and vertical (y) distance. For example, if we have values as 2 along the horizontal axis and 5 along the vertical axis, then the representation will be (2,3). It has to be noted that the first value will always be from the x-axis.
(Image will be uploaded soon)
How to Define Cartesian Coordinates on a Plot?
Consider a point on an x-y coordinate system to determine the cartesian coordinates of a point.
First of all, calculate how far from the origin it is along the x-axis, i.e. its perpendicular distance from the y-axis which gives an x-coordinate value.
Next measure how far, in a perpendicular direction from the x-axis, the point is along the y-axis. This provides the y-coordinate value.
Negative Values of x and y in the Cartesian Coordinate System
We also have negative values in the Cartesian coordinate system, much like the Number Line system.
Except in the first coordinate system, all coordinates have negative values either in x-axis or y-axis or both axis.
Here in the 1st coordinate system, both the x and y-axis values are positive.
The 2nd coordinate system has negative x value and positive y value.
The 3rd coordinate system has both x and y coordinate values as negative.
The 4th coordinate system has positive x value and negative y value.
(Image will be uploaded soon)
Cartesian Coordinates in Three-Dimensions
The position of the point in space must be described by three coordinates in a three-dimensional Cartesian coordinate system, usually three coordinates (x, y, z). The point is somewhere on a flat plane inside a two-dimensional structure. A plane, however, has only a length and a width, whereas a three-dimensional space must have a height or a depth as well. We can think of the point in this case as being somewhere inside a rectangular box.
As in a two-dimensional system, the first two coordinates, x and y, are defined in the same way. These define the direction of the point if it was projected onto the x-y plane downwards (or upwards) at right angles. For example, Imagine holding a ball in your hand, arm outstretched at shoulder height, now drop the ball. It bounces on the deck, directly below the location of your side. The point at which the ball bounces is the (x,y) coordinate of the point if the field is the x-y plane.
There is also z-axis to the three-dimensional system, which lies perpendicular to the x-y plane. The location above the ground where the ball was held in its z-coordinate. The point at which x, y and z are all equal to zero is the origin of the three-dimensional Cartesian system (0,0,0).
Basic Examples Regarding Cartesian Coordinate Systems
1) If the abscissa is 3 and ordinate is 7. How is the cartesian coordinate system represented?
Ans: Here abscissa means the horizontal axis (x) value and ordinate mean vertical axis (y) value. So we get the coordinate system value as (3,7). The below figure gives the location of the points on the cartesian coordinate system.
(Image will be uploaded soon)
2) How to plot the values of abscissa and ordinate on the cartesian coordinate system?
x = 10, y = -5
x = -20, y = 10
Ans: For the first points x = 10 and y = -5, move 10 units along the positive x-axis and 5 units along the negative y-axis. In the given plot below the line represented in green and orange colour represents this plot.
Now for the second points x = -20 and y = 10, move 20 units along the negative x-axis and 10 units along the positive y-axis. In the given plot below the line represented in purple and black colour represents this plot.
(Image will be uploaded soon)
Applications of Cartesian Coordinate Systems
Cartesian coordinates are the basis of analytic geometry and provide many other branches of mathematics with enlightening geometric meanings, such as linear algebra, complicated analysis, differential geometry, multivariate calculus, group theory, and more.
For most applied disciplines that deal with geometry, including astronomy, physics, engineering and many more, Cartesian coordinates are also important tools.
They are the most common coordinate system used in computer graphics, geometric design supported by computers, and other data processing related to geometry.
FAQs on Cartesian Coordinates Explained: Plots, Rules & Examples
1. What are Cartesian coordinates?
The Cartesian coordinate system is a fundamental mathematical tool used to uniquely determine the position of a point on a plane. It consists of two perpendicular number lines, the horizontal x-axis and the vertical y-axis, that intersect at a point called the origin (0,0). Any point is located using an ordered pair of numbers, (x, y). This system is also commonly known as the Rectangular Coordinate System.
2. How do you write the coordinates of a point in the Cartesian plane?
The coordinates of a point are written as an ordered pair (x, y). The first number, 'x', is the abscissa, which represents the signed horizontal distance from the origin along the x-axis. The second number, 'y', is the ordinate, which represents the signed vertical distance from the origin along the y-axis. For example, the point P(4, -2) is 4 units to the right of the origin and 2 units down.
3. What is the difference between an abscissa and an ordinate?
The abscissa and ordinate are the two components of a Cartesian coordinate. The abscissa is the x-coordinate, representing the point's horizontal position relative to the y-axis. The ordinate is the y-coordinate, representing the point's vertical position relative to the x-axis. Together, the pair (abscissa, ordinate) or (x, y) precisely locates the point.
4. What are the four quadrants of a Cartesian plane and their sign conventions?
The x-axis and y-axis divide the plane into four regions called quadrants, numbered counter-clockwise from the top right. The sign conventions for a coordinate (x, y) in each quadrant are as follows:
- Quadrant I: Both x and y are positive (+, +).
- Quadrant II: x is negative, and y is positive (–, +).
- Quadrant III: Both x and y are negative (–, –).
- Quadrant IV: x is positive, and y is negative (+, –).
5. How do you plot a point, for instance, P(-3, 5), on a Cartesian plane?
To plot the point P(-3, 5), you follow these steps:
1. Start at the origin (0,0).
2. Look at the x-coordinate (-3). Since it is negative, move 3 units to the left along the x-axis.
3. From that position, look at the y-coordinate (5). Since it is positive, move 5 units upwards, parallel to the y-axis.
4. Mark this final position as the point P(-3, 5). The point will be in Quadrant II.
6. Why is the origin (0, 0) so important in the Cartesian coordinate system?
The origin (0, 0) is the most crucial reference point in the system. It serves as the universal starting point for all measurements. Both the abscissa (x-value) and the ordinate (y-value) of any point are defined by their distance and direction from the origin. Without the origin as a fixed, zero-reference anchor, there would be no consistent way to define the location of any other point on the plane.
7. How does the Cartesian coordinate system extend from two to three dimensions?
The system extends to three dimensions by adding a third axis, the z-axis, which is perpendicular to both the x-axis and y-axis. This creates a 3D space where the location of any point is defined by an ordered triple (x, y, z). Each value represents the point's signed distance from the origin along the respective axis, allowing for the precise location of objects in space, not just on a flat plane.
8. What is the fundamental difference between Cartesian and Polar coordinates?
The fundamental difference lies in how they define a point's location.
- Cartesian coordinates use two perpendicular distances (x and y) from two fixed axes. It's like navigating a city grid: "go 3 blocks east and 4 blocks north".
- Polar coordinates use a single distance from the origin (radius, r) and an angle (θ) from a fixed direction. It's like giving a direction: "go 5 km in a 30-degree direction".
9. What is the formula to find the distance between two points in a Cartesian plane?
To find the distance 'd' between two points, A(x₁, y₁) and B(x₂, y₂), you use the distance formula, which is derived from the Pythagorean theorem. The formula is:
d = √[(x₂ – x₁)² + (y₂ – y₁)²]
This calculates the length of the straight line segment connecting the two points.
10. How are Cartesian coordinates used in real-world applications like computer graphics?
Cartesian coordinates are essential in many real-world fields. In computer graphics and video games, the screen is a 2D coordinate plane where every pixel has an (x, y) address. For 3D models and game environments, a 3D (x, y, z) system is used to define the position, size, and orientation of every object, character, and camera in the virtual space. This allows for precise rendering and interaction.
