Cartesian Coordinates

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.

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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.

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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.

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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.

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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.