Representation of points in a plane

Representation of points in a plane

We are familiar with the representation of real numbers on a line, which we call a real line. In this representation we fix a point O (called origin) and represent a real number by a point A on this line such that its distance OA (see figure given below) is equal to the value of real number. The left side of O represents negative real numbers and the right side of O represents positive real numbers. Thus, not only the magnitude of OA but the direction of the line OA is also considered for representation. Here  OA= -AO

Similarly ordered pairs are represented in a plane. To represent an ordered pair (a, b) we take two reference lines which are mutually perpendicular. The ordered pair (a, b) represents in such a plane, by a point P(a, b) such that (see figure given below) OA = a and OB = b.

This system is called Cartesian co-ordinate system. Since elements of an ordered pair are not inter changeable (i.e., (a, b) ≠ (b, a) unless a = b) so they are represented in particular order, the first element ‘a’ is represented on horizontal line called abscissa and the second element ‘b’ on a vertical line called ordinate. Like the real number notation the positive side of the x-axis is the right side of O and positive side of y-axis is upper side of O.

  

So, the two lines divide the region in 4 parts. These are called quadrants. These quadrants are characterized as

 I quadrant     x > 0, y > 0

II quadrant     x < 0, y > 0

III quadrant    x < 0, y < 0

IV quadrant    x > 0, y < 0

 

Here the point ‘O’ represents x = 0 and y = 0, hence order pair becomes (0, 0).

There is a second type of representation called the polar co-ordinate system. In this system a reference is fixed to a line (Called the initial line), and a point called the origin in the system. Any point P is represented by ordered pair (r, θ). 

Such that

OP = r; the distance of point from origin.

and ∠POX = θ The angular displacement of line OP from fixed line i.e. the initial line, (in the anticlockwise direction)

Clearly ‘a’ = r cos θ and ‘b’ = r sin θ