Cartesian Coordinate Systems

1. The Cartesian Coordinate Systems are Orthogonal Coordinate Systems, having all the Coordinate Axes that are Straight Lines.
2. An \(N\) Dimensional Cartesian Coordinate System has \(N\) Number of Mutually Perpendicular Straight Line Coordinate Axes that is used to represent an \(N\)- Dimensional Space. The 1, 2 and 3 Dimensional Cartesian Coordinate Systems are used most frequently.
3. A 1 Dimensional Cartesian Coordinate System consists of a Single Coordinate Axis called the Real Number Line.
   
   This Real Number Line extends from Location \(-\hspace{1mm}\infty\) on the Left to Location \(+\hspace{1mm}\infty\) on the Right.
   
   
   
   Every Distinct Location on this Line is represented by a Unique Real Number denoted by Variable \(x\).
   
   The Origin Location \(0\) lies in the Middle of the Number Line.
4. A 2 Dimensional Cartesian Coordinate System consists of a Perpendicular Set of Horizontal and Vertical Coordinate Axes.
   
   The Horizontal Axis called the \(X\) Axis extends from Location \(-\hspace{1mm}\infty\) on the Left to Location \(+\hspace{1mm}\infty\) on the Right.
   
   The Vertical Axis called the \(Y\) Axis extends from Location \(-\hspace{1mm}\infty\) on the Bottom to Location \(+\hspace{1mm}\infty\) on the Top.
   
   
   
   The Grid for the 2 Dimensional Cartesian Coordinate System is formed by the Intersections of the Straight Lines Parallel to \(X\) Axis and the Straight Lines Parallel to \(Y\) Axis.
   
   Each such Point of Intersection specifies a Distinct Location on the Grid of the Plane represented by a 2 Dimensional Cartesian Coordinate System known as a Coordinate Location or a Coordinate Point and is given by a Unique Ordered Pair of Real Number Values \((x,y)\).
   
   The First Value of the Ordered Pair, (i.e. \(x\)), gives the Value of \(X\) Axis for the Location (also called \(X\) Coordinate).
   
   The Second Value of the Ordered Pair, (i.e. \(y\)), gives the Value of \(Y\) Axis for the Location (also called \(Y\) Coordinate).
   
   The Intersection of the 2 Coordinate Axes forms the Origin of the 2 Dimensional Cartesian Coordinate System having Coordinate Location \((0,0)\).
5. A 3 Dimensional Cartesian Coordinate System consists of a 3 Mutually Perpendicular Set of Coordinate Axes, the Horizontal Axis, the Vertical Axis and the Front-Back Axis.
   
   The Horizontal Axis extends from Location \(-\hspace{1mm}\infty\) on the Left to Location \(+\hspace{1mm}\infty\) on the Right.
   
   The Vertical Axis extends from Location \(-\hspace{1mm}\infty\) on the Bottom to Location \(+\hspace{1mm}\infty\) on the Top.
   
   The Front-Back Axis extends from Location \(-\hspace{1mm}\infty\) on the Back to Location \(+\hspace{1mm}\infty\) on the Front.
   
   These Coordinate Axes are also labelled as \(X\), \(Y\), and \(Z\) Axes.
   
   However, unlike 2 Dimensional Cartesian Coordinate System where the Directions of \(X\) Axis (Horizontal) and \(Y\) Axis (Vertical) are fixed, in 3 Dimensional Cartesian Coordinate System the Assignment of Directions to \(X\), \(Y\), and \(Z\) is Not Fixed but rather depend on whether the Coordinate System is Right Handed or Left Handed.
   
   Figure (1) below gives the 3 possible orientations of Right Handed 3 Dimensional Cartesian Coordinate Sytems.
   
   
   
   A 3 Dimensional Cartesian Coordinate Sytem is Right Handed if its \(X\), \(Y\) and \(Z\) Coordinate Axes are aligned such that they satisfy following 3 conditions
   1. 90 \(^\circ\)Counter-Clockwise Rotation along the Plane defined by \(X\) and \(Y\) Axes of a Point lying on \(X\) Axis in its Positive Direction transforms the Point to the \(Y\) Axis in its Positive Direction.
   2. 90 \(^\circ\)Counter-Clockwise Rotation along the Plane defined by \(Y\) and \(Z\) Axes of a Point lying on \(Y\) Axis in its Positive Direction transforms the Point to the \(Z\) Axis in its Positive Direction.
   3. 90 \(^\circ\)Counter-Clockwise Rotation along the Plane defined by \(Z\) and \(X\) Axes of a Point lying on \(Z\) Axis in its Positive Direction transforms the Point to the \(X\) Axis in its Positive Direction.
   The above 3 conditions can be summarised as Right Hand Thumb Rule which states that if we Curl the Fingers of our Right Hand while Sticking the Thumb out, the curling of the fingers denotes the 90 \(^\circ\) Counter-Clockwise Rotation of a Point from Positive Direction of 1st Axis to Positive Direction of 2nd Axis, where as the Thumb gives the Direction of the Positive Direction of the 3rd Axis, where in the 1st, 2nd and 3rd Axis can be assigned to \(X\), \(Y\) and \(Z\) Axis in a Cyclic Manner (i.e [\(X\hspace{1mm}Y\hspace{1mm}Z\)], [\(Y\hspace{1mm}Z\hspace{1mm}X\)] or [\(Z\hspace{1mm}X\hspace{1mm}Y\)]).
   
   Figure (2) below gives the 3 possible orientations of Left Handed 3 Dimensional Cartesian Coordinate Sytems.
   
   
   
   A 3 Dimensional Cartesian Coordinate Sytem is Left Handed if its \(X\), \(Y\) and \(Z\) Coordinate Axes are aligned such that they satisfy following 3 conditions
   1. 90 \(^\circ\) Clockwise Rotation along the Plane defined by \(X\) and \(Y\) Axes of a Point lying on \(X\) Axis in its Positive Direction transforms the Point to the \(Y\) Axis in its Positive Direction.
   2. 90 \(^\circ\) Clockwise Rotation along the Plane defined by \(Y\) and \(Z\) Axes of a Point lying on \(Y\) Axis in its Positive Direction transforms the Point to the \(Z\) Axis in its Positive Direction.
   3. 90 \(^\circ\) Clockwise Rotation along the Plane defined by \(Z\) and \(X\) Axes of a Point lying on \(Z\) Axis in its Positive Direction transforms the Point to the \(X\) Axis in its Positive Direction.
   The above 3 conditions can be summarised as Left Hand Thumb Rule which states that if we Curl the Fingers of our Left Hand while Sticking the Thumb out, the curling of the fingers denotes the 90 \(^\circ\) Clockwise Rotation of a Point from Positive Direction of 1st Axis to Positive Direction of 2nd Axis, where as the Thumb gives the Direction of the Positive Direction of the 3rd Axis, where in the 1st, 2nd and 3rd Axis can be assigned to \(X\), \(Y\) and \(Z\) Axis in a Cyclic Manner (i.e [\(X\hspace{1mm}Y\hspace{1mm}Z\)], [\(Y\hspace{1mm}Z\hspace{1mm}X\)] or [\(Z\hspace{1mm}X\hspace{1mm}Y\)]).
   
   The Grid for the 3 Dimensional Cartesian Coordinate System is formed by the Intersections of the Straight Lines Parallel to \(X\) Axis, the Straight Lines Parallel to \(Y\) Axis and the Straight Lines Parallel to \(Z\) Axis.
   
   Each such Point of Triple Intersection specifies a Distinct Location on the Grid of the 3 Dimensional Space represented by a 3 Dimensional Cartesian Coordinate System known as a Coordinate Location or a Coordinate Point and is given by a Unique Ordered Set of Real Number Values \((x,y,z)\).
   
   The First Value of the Ordered Set, (i.e. \(x\)), gives the Value of \(X\) Axis for the Location (also called \(X\) Coordinate).
   
   The Second Value of the Ordered Set, (i.e. \(y\)), gives the Value of \(Y\) Axis for the Location (also called \(Y\) Coordinate).
   
   The Third Value of the Ordered Set, (i.e. \(z\)), gives the Value of \(Z\) Axis for the Location (also called \(Z\) Coordinate).
   
   The Intersection of the 3 Coordinate Axes forms the Origin of the 3 Dimensional Cartesian Coordinate System having Coordinate Location \((0,0,0)\).