Cartesian Coordinate System is used to describe the position of a point in a plane using coordinates in coordinate geometry. Coordinates are the way of telling the position of a point using numbers.
Rene Discartes invented the Coordinate System and the term Cartesian Coordinate System is named after the name of Discartes mathematician to honour the inventor.
The cartesian geometry system introduces coordinate axes, coordinates and quadrants to describe the position of a point in a plane and its distances from other points.
x and y coordinate axes
The cartesian coordinate system uses a plane which has two mutually perpendicular lines which are called coordinate axes. Out of the two perpendicular lines, one line is in horizontal position and another line is in vertical position. The figure below shows how the two coordinate axes look like.
The horizontal line marked with XX' is called the x-axis.
The vertical line marked with YY' is called the y-axis.
Mind the words axis and axes, how they are used. Axis is used to represent one axis, it is singular. On the other hand, axes is used to represent more than one axis, which is plural.
Positive and negative directions of axes
In the figure below, both axes, x-axis and y-axis intersect each other at point O and are perpendicular to each other. The point O is called the origin.
The x-axis has two sides, one side lies on the right side of the y-axis and can be read as OX. The second side lies on the left side of the y-axis and is read as OX'. Similarly, the y-axis has two sides, top and bottom of the x-axis. The top side of the y-axis is read as OY and bottom side of the y-axis is OY'.
The OX on x-axis and OY on y-axis are called as positive directions of x-axis and y-axis respectively. Similarly, the OX' on x-axis and OY' on y-axis are called negative directions of x-axis and y-axis respectively.
Positive directions of x-axis = OX
Positive directions of y-axis = OY
Negative directions of x-axis = OX'
Negative directions of y-axis = OY'
How to scale x and y axes?
Scaling of axes is meant by marking the x-axis and y-axis with numbers which are placed at equal distances.
In the above figure, the ray OX on x-axis has positive numbers (1, 2, 3, 4, ....) and ray OX' has negative numbers (-1, -2, -3, -4, ....) only.
Similarly, The ray OY on the y-axis has positive numbers (1, 2, 3, 4, ....) and ray OY' has negative numbers (-1, -2, -3, -4, ....) on it. In other words, we can say that positive numbers lie in the directions of ray OX and ray OY and the negative numbers lie in the direction of ray OX' and ray OY'.
What are the four quadrants?
The two coordinate axes x-axis and y-axis in coordinate geometry divide the plane into four parts. These four parts are called quadrants.
Here, the x-axis and y-axis is divide the plane into four parts which are XOY, X'OY, X'OY' and XOY'.
XOY is called the first quadrant.
X'OY is called the second quadrant.
X'OY' is called the third quadrant.
XOY' is called the fourth quadrant.
Also, the four quadrants are numbered as I, II, III and IV anticlockwise starting from the first quadrant XOY and the last quadrant as XOY'.
Locate coordinates of a point
Any point in a plane can lie in any of the four quadrants I, II, III or IV. The position of a point in the plane is called the coordinates of that point. Coordinates of the point in a quadrant is determined by knowing its perpendicular distances from its nearest x-axis and y-axis.
Coordinates of a point is written by writing its perpendicular distance in brackets in the format of (x,y), where x is the perpendicular distance of the point from x-axis and y is the perpendicular distance of the point from y-axis.
The x-axis coordinate of a point is called an abscissa.
The y-axis coordinate of a point is called ordinate.
Coordinates of a point can be written only in specific order i.e (x,y). First we write abscissa followed by ordinate and separated by a comma.
In the above figure, P is the point that lies in the first quadrant and is marked as P.
The perpendicular distance PN of point P from y-axis is 3 units, measured along the positive direction of x-axis as OM which is 3 units.
The perpendicular distance PM of point P from x-axis is 4 units, measured along the positive direction of y-axis as ON which is 4 units.
So, the perpendicular distance of point P from y-axis is 3 and x-axis is 4, so the coordinate of point P can be written as (3,4).
Abscissa and ordinate of a point cannot be interchanged.
i.e. (x,y) ≠ (y,x)
Moreover, (x,y) = (y,x) if x = y
Coordinates of origin are always written as (0,0) because the perpendicular distance of origin from x-axis is zero and from y-axis is also zero.
The ordinate of any point on the x-axis is 0 i.e coordinate of any point on the x-axis is (x,0).
The abscissa of any point on the y-axis is 0 i.e coordinate of any point on the y-axis is (0,y).
Sign conventions used in quadrants
Any point that lies in the first quadrant will always have positive abscissa and positive ordinate. The first quadrant has points with sign (+,+).
The point that lies in the second quadrant has negative abscissa and positive ordinate. The second quadrant has points with sign (-,+).
Any point that lies in the third quadrant has negative abscissa and negative ordinate. The third quadrant has points with sign (-,-).
Any point that lies in the fourth quadrant has positive abscissa and negative ordinate. The fourth quadrant has points with sign (+,-).
X-axis XOX' and y-axis YOY' divide the coordinate plane into four quadrants.
The ray OX is regarded as a positive x-axis.
The ray OY is regarded as a positive y-axis.
The ray OX' is regarded as the negative x-axis.
The ray OY' is regarded as the negative y-axis.
