Number line:
In the above number line, the position of point \(P\) in the above line can be represented as \(6\) units with reference to one line (horizontal line).
A similar rule applies to the negative side of the number line as well. It is possible to represent the position of a point with reference to more than one point.
Descartes invented placing two such number lines perpendicular to each other on a plane and locating points on the plane by referring them to these lines. The perpendicular lines may be in any direction. But, when we choose these two lines to locate a point in a plane in this chapter, one line will be horizontal(First image) and the other will be vertical(Second image), as below:
Descartes invented placing two such number lines perpendicular to each other on a plane and locating points on the plane by referring them to these lines. The perpendicular lines may be in any direction. But, when we choose these two lines to locate a point in a plane in this chapter, one line will be horizontal(First image) and the other will be vertical(Second image), as below:
Let us combine these \(2\) lines in such a way they are perpendicular to each other. These two lines intersect each other at the point \((0,0)\).
Cartesian system:
- It is a co-ordinate number system used to describe the position of a point in two dimensions by means of two perpendicular lines (x axis and y axis).
- The line \(X'OX\) is the horizontal line called the \(x\) - axis.
- The line \(Y'OY\) is the vertical line called the \(y\) - axis.
Signs in the graphs:
- In \(x\) - axis, the point is positive along the direction \(OX\) and negative along the direction \(OX'\).
- In \(y\) - axis, the point is positive along the direction \(OY\) and negative along the direction \(OY'\).
Co-ordinate axes:
The plural form of the axis is called the axes. A number line represented horizontally is the \(x\) - axis, and a number line represented vertically is the \(y\) - axis. Joining both the lines together at origin is called the \(xy\) plane or cartesian plane or co-ordinate axes.
The \(x\) - axis and \(y\) - axis divide the cartesian plane into four regions from the origin. These are called quadrants. They are usually numbered in anticlockwise direction starting from the region bounded by positive \(x\) and \(y\) axis (that is \(OX\)).
Quadrant I:
- Any point located in quadrant \(I\) will have a positive number in the \(x\) - axis and \(y\) - axis.
- That is, \(x > 0\), \(y > 0\).
- The region is \(XOY\).
Example:\((2,3)\), \((6,10)\), \((9,12)\)
Quadrant II:
- Any point located in quadrant \(II\) will have a negative number in the \(x\) - axis and a positive number in \(y\) - axis.
- That is, \(x < 0\), \(y > 0\).
- The region is \(X'OY\).
Example:\((-3,6)\), \((-2,5)\), \((-15,12)\)
Quadrant III:
- Any point located in quadrant \(III\) will have a negative number in the \(x\) - axis and \(y\) - axis.
- That is, \(x < 0\), \(y < 0\).
- The region is \(X'OY'\).
Example:\((-5,-6)\), \((-2,-1)\), \((-8,-10)\)
Quadrant IV:
- Any point located in quadrant \(IV\) will have a positive number in the \(x\) - axis and a negative number in \(y\) - axis.
- That is, \(x > 0\), \(y < 0\).
- The region is \(XOY'\).
Example:\((1,-3)\), \((3, -4)\), \((7,-1)\)
Coordinate of a point on the axes:
- If a point lies on the \(x\) - axis, then the coordinate is \((x,0)\). That is, \(y=0\).
Example:\((-3,0)\), \((6,0)\), \((1,0)\).
- If a point lies on the \(y\) - axis, then the coordinate is \((0,y)\). That is, \(x=0\).
Example:\((0,-1)\), \((0,5)\), \((0,10)\).