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Cartesian Coordinate

The invention of Cartesian coordinates in the 17th century by René Descartes revolutionized mathematics by providing the first systematic link between Euclidean geometry and algebra.

A coordinate system consists of four basic elements:
Choice of origin
Choice of axes
Choice of positive direction for each ax
Choice of unit vectors for each axes


Infinitesimal Line Element:
Consider a small infinitesimal displacement ds between two points P1 and P2 (Figure1). In Cartesian coordinates this vector can be decomposed into:


Figure 1: Displacement between two points.

Infinitesimal Area Element:
An infinitesimal area element of the surface of a small cube is given by
dA= (dx) (dy)
Area elements are actually vectors where the direction of the vector  is perpendicular to the plane defined by the area. Since there is a choice of direction, we shall choose the area vector to always point outwards from a closed surface, defined by the right-hand rule. So for the above, the infinitesimal area vector is:


Infinitesimal Volume Element:
An infinitesimal volume element in Cartesian coordinates is given by
dV = dx dy dz

The position of any point on the Cartesian plane is described by using two numbers:  (x, y). The first number x is the horizontal position of the point from the origin.  It is called the x-coordinate. The second number, y, is the vertical position of the point from the origin.  It is called the y-coordinate.  Since a specific order is used to represent the coordinates, they are called ordered pairs



For example, the ordered pair (5, 9) represents point 5 units to the right of the origin in the direction of the x-axis and 9 units above the origin in the direction of the y-axis.

In a system formed by a point,
O, and an orthonormal basis at each point, P, there is a corresponding vector  in the plane such that:


The coefficients x and y of the linear combination are called coordinates of point P.
The first x is the abscissa and the second y is the ordinate. As the linear combination is unique, each point corresponds to a pair of numbers and each pair of numbers to a point.

Using Cartesian coordinates on the plane, the distance between two points (
x1y1) and (x2y2) is defined by the formula
,
which can be viewed as a version of the Pythagorean Theorem. Similarly, the angle that a line makes with the horizontal can be defined by the formula

θ = tan-1(m),
where m is the slope of the line.

Midpoint formula
If (x1,y1) and (x2,y2) are two points, the midpoint of the line joining these two points is given by:

Example 1: Plot the graph of Cartesian coordinate points A (2, − 4), B (4, 3), C (−2, 3) and D (−3, −4).
Solution: 
As shown in the figure:
A (2, − 4) lies in the fourth quadrant,
B (4, 3)  lies in the first quadrant,
C (−2, 3) lies in the second quadrant,
D (−3, −4) lies in the third quadrant.




Example 2: Find the Cartesian coordinate distance between the points A (3, 6) and B (6, 3).
Solution:
Let "d" be the distance between A and B.        

(x1,y1) = (3, 6) and (x2,y2) = (6,3)
Then d (A, B) is determined by:


 

Example 3: Find the equation of the line having slope 7 and y-intercept -3
Solution:
Applying the slope-intercept formula, the equation of the line is given by: y = mx + c, where m = 7, and c = -3.

Insert the values we have:
y = 7x + (-3)
y = 7x - 3
Rearranging the above equation, we get
7x - y - 3 = 0.


Example 4: Determine the midpoint coordinates of a line segment joining points A (6, 8) and B (-1,-5)
Solution:
We know that:

x1 = 6
x2 = -1
y1 = 8
y2 = -5

The required mid point is given by:

Let the midpoint be denoted by M (a, b), we have:


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