Cartesian System

A plane formed by two number lines, one horizontal

and the other vertical, such that they intersect each

other at their zeroes, and then they form a Cartesian

Plane.

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● The horizontal line is known as the x-axis and the vertical line is known

as the y-axis.

● The point where these two lines intersect each other is called the origin.

It is represented as ‘O’.

● OX and OY are the positive directions as the positive numbers lie in the

right and upward direction.

● Similarly, the left and the downward directions are the negative directions

as all the negative numbers lie there.

Quadrants of the Cartesian Plane The Cartesian plane is divided into four quadrants namely Quadrant I, Quadrant II, Quadrant III, and Quadrant IV anticlockwise from OX.

Coordinates of a Point To write the coordinates of a point we need to follow the following rules. ● Thex – coordinate of a point is marked by drawing perpendicular from the y-axis measured a length of the x-axis .It is also called the Abscissa.

They – coordinate of a point is marked by drawing a perpendicular from the x-axis measured a length of the y-axis .It is also called the Ordinate. ● While writing the coordinates of a point in the coordinate plane, the x – coordinate comes first, and then the y – coordinate. We write the coordinates in brackets. In figure, OB = CA = x coordinate (Abscissa), and CO = AB = y coordinate (Ordinate). We write the coordinate as (x, y).

Remarks:

As the origin, O has zero distance from the x-axis and the y-axis so its abscissa and ordinate are zero. Hence the coordinate of the origin is (0, 0). The relationship between the signs of the coordinates of a point and the quadrant of a point in which it lies.

Plotting a Point in the Plane if its Coordinates are Given

Steps to plot the point (2, 3) on the Cartesian plane:

● First of all, we need to draw the Cartesian plane by drawing the coordinate axes with 1 unit = 1 cm.

● To mark the x coordinates, starting from 0 moves towards the positive x-axis and counts to 2.

● To mark the y coordinate, starting from 2 moves upwards in the positive direction and counts to 3.

● Now this point is the coordinate (2, 3). Likewise, we can plot all the other points, like (-3, 1) and (-1.5,-2.5) in the figure.

Question: Are the coordinates (x, y) = (y, x)? Let x = (-4) and y = (-2) So (x, y) = (- 4, – 2) (y, x) = (- 2, – 4)

Let’s mark these coordinates on the Cartesian plane. You can see that the positions of both the points are different in the Cartesian plane. So, If x ≠ y, then (x, y) ≠ (y, x), and (x, y) = (y, x), if x = y.

Example: Plot the points (6, 4), (- 6,- 4), (- 6, 4) and (6,- 4) on the Cartesian plane.

Solution: Since both numbers 6, 4 are positive the point (6, 4) lies in the first quadrant. For x coordinate, we will move towards the right and count to 6. Then from that point go upward and count to 4. Mark that point as the coordinate (6, 4). Similarly, we can plot all the other three points.