Cartesian Plane

What is Cartesian Plane?

A Cartesian plane is a graph with two axes, one is called the x-axis and the other one is the y-axis. These two axes are perpendicular to each other. The origin (O) is in the exact center of the graph intersecting point of the two axes. Numbers to the right of the zero on the x-axis are positive and the numbers to the left of zero are negative. For the y-axis, numbers below zero are negative and numbers above are positive.Let us study What is cartesian plane in detail.The Cartesian Plane is also referred to as the x-y plane or the coordinate plane.

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Cartesian Plane Definition

A graph with two perpendicular number lines used to describe any point in the plane using an ordered pair of numbers is called a Cartesian Plane or the coordinate plane. Where one is vertically referred to an y -axis and the other is horizontally referred to as  x- axis and both form right angles with one another.

The Great Mathematician Rene Descartes Latin name was Renatius Cartesius,who originally came up with the concept Cartesian plane named it after him. Points on the cartesian plane are called ordered pairs written as (x , y).

The values that come first in ordered pairs are plotted along the horizontal line(x-axis) while the second number of ordered pairs is plotted along the vertical line(y-axis). 


Cartesian Plane Quadrants

Take a plain sheet of paper, draw one horizontal line called X-axis and one vertical line called Y-axis such that it intersects at a point O. These two axes divide the cartesian plane into 4 parts each part is called a quadrant. A quadrant is ¼th part of a cartesian plane divided by coordinate axes.

There are four corners you need to know about on the graph.

  • First Quadrant  = Top right.

  • Second Quadrant  = Top left.

  • Third Quadrant  = Bottom left.

  • Fourth Quadrant  = Bottom right.

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In the first quadrant, both the coordinates are positive.

In the second quadrant, x-coordinate is negative whereas the y-coordinate is positive

In the third quadrant, both the coordinates are negative.

In the fourth quadrant, the x-coordinate is positive whereas the y-coordinate is negative.


Ordinate and Abscissa Definition

Ordered pairs are called the ordinate and abscissa on a Cartesian plane.


Abscissa

The distance of any point on the plane from the Y-axis is called the abscissa.The abscissa is the first number in an ordered pair. It is the x-value.


Ordinate

The distance of any point on the plane from the X-axis is said to be ordinate.

The ordinate the second number in an ordered pair. It is the y-value.

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Origin:

The center point of intersection from which the distances are marked is called the origin. In a two- dimensional plane, the X-axis, and Y-axis intersect at a point is called the origin.

The above graph shows four ordered pairs:

  • The first point (-3,1), -3 is the abscissa and 1 is the ordinate.

  • The second point (2,3), 2 is the abscissa and 3 is the ordinate.

  • The point (0,0), 0 is the abscissa and the ordinate.

  • For the point (-1.5,-2.5), -1.5 is the abscissa and -2.5 is the ordinate.

Don’t forget that the first point you get will always be on the X portion of the graph, and the second number will always be the Y.


One Dimensional Cartesian Plane

In one dimensional Cartesian Plane, draw a straight line and choose a point O as the origin in the center of the line. Then write numbers, positive 1 to infinity to the right and negative 1 to infinity to the left. This line is also referred to as a number line. Here any point can have only single number position hence it is called as one-dimensional cartesian plane

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Two Dimensional Cartesian Plane

A two-dimensional plane that is formed by the intersection of one horizontal number line and one vertical number line, referred as x-axis and y-axis respectively is called a two dimensional cartesian plane. These lines are perpendicular to each other and intersect at a point O called the origin. These axes divide the plane into four parts called quadrants. This plane is called the cartesian plane or coordinate plane or XY- plane.

For  any given point P, let x and y be the corresponding number lines and the coordinates are written as (x, y), where x is called the abscissa, and y is called the ordinate.

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Three Dimensional Cartesian Plane

The three-dimensional Cartesian plane has one more axis perpendicular to the normal Cartesian plane. There is a third axis, Z-axis which is perpendicular to the XY plane. Any point lying on this plane is defined by the set of three points (x, y, z). Here, x defines the position along the X-axis, y defines the position along the Y-axis, and z defines the position along the Z-axis. The given graph shows the three dimensional Cartesian plane with three axes X, Y, and Z.

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Solved Examples

Example 1: Plotting points (1,2) on a cartesian board.

Solution: To plot a point, move along the x axis to find the first coordinate (the first number), then move up or down to find the second coordinate.

The first point you get will always be on the X portion of the graph, and the second number will always be the Y. For example, (1,2) means 1 space1 on the x-axis and 2 on the y-axis.

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Example 2: Plot A(2,-2) , B(-5,3), and C(-4,-6)

Solution:

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Quiz Time

  1. Plot the following points on the cartesian board.

    1. (-4, 4)

    2. (2, 0)

    3. (5, -2)

    4. (3, -3)

FAQ (Frequently Asked Questions)

1. What is an Origin?

The origin is the reference point used to define all other points; a fixed point referred to when noting the geometry of a space. With this in mind, it is often denoted by O, and the coordinates are always zero.

In one dimension we simply write the origin as 0, it’s the starting point where we start numbering on a number line. You can go in either of two directions:

  • Going left, you would count negative numbers infinitely.

  • Going right, you would count positive numbers infinitely.

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In two dimensions, using the Cartesian plane, an origin is the point where the x and y axes intersect. This point is written as (0, 0).

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In three dimensions, the origin is (0, 0, 0) and is defined as the place the x, y, and z axis intersect.

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2. What are the Uses of Cartesian Planes

Cartesian planes are used to plot the solutions to formulas with two variables, typically represented by x and y, though other symbols can be substituted for the x- and y-axis, so long as they are properly labeled and follow the same rules as x and y in the function.

These visual tools provide students with a pinpoint using these two points that account for the solution to the equation.

Students may also solve for a missing variable if x or y is unknown by simplifying the equation until both variables have a solution and can be plotted on a Cartesian plane. This process forms the basis for most early algebraic computations and data mapping.