Linear Programming

Introduction to Three Dimensional Geometry
Basics
Cartersian plane

• A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a
plane by a pair of numerical coordinates, which are the signed distances to the point from two
fixed perpendicular directed lines, measured in the same unit of length.
Coordinates of a Point

• The coordinates of a point are a pair of numbers that define its exact location on a two-
dimensional plane
• A number on the x-axis called an x-coordinate, and a number on the y-axis called a y-
coordinate.
Ordered Pair

• An ordered pair contains the coordinates of one point in the coordinate system.
• The order in which you write x- and y-coordinates in an ordered pair is very important.
• The x-coordinate always comes first, followed by the y-coordinate
• There is also a three coordinate called Z coordinate
One-Dimensional Coordinate Systems

• A 1-dimensional coordinate system, also known as a number line, uses one coordinate to tell
how far away from zero it is located.




Two-Dimensional Coordinate Systems

• The below shown figure represens the Co-ordinate plane or it is also called as Cartersian
plane OR XY-Plane
• Each reference line is called a coordinate axis or just axis (plural axes) of the system
• The coordinates are written as an ordered pair (x, y).
• The point where the axes meet is the common origin of the two number lines and is simply called
the origin.
• It is often labelled O and if so then the axes are called Ox and Oy.
• The value of x is called the x-coordinate or abscissa
• The value of y is called the y-coordinate or ordinate




Three Dimensional Coordinate Axis

, • The three straight lines having a common point that are the intersections of the three coordinate
planes of reference in three-dimensional Cartesian geometry
• Three-dimensional space by means of three coordinates.
• Three coordinate axes are given, each perpendicular to the other two at the origin, the point at
which they cross. They are usually labelled x, y, and z.
• The position of any point in three-dimensional space is given by an ordered triple of real
numbers, each number giving the distance of that point from the origin measured along the
given axis, which is equal to the distance of that point from the plane determined by the other
two axes
• The coordinates are written as an ordered pair (x, y, z).




Coordinate Axes and Coordinate Planes in the Three Dimensional Space
• Consider three planes intersecting at a point O such that these three planes are mutually
perpendicular to each other
• Three planes intersect along the lines X′OX, Y′OY and Z′OZ, called the x, y and z-axes
• The planes XOY, YOZ and ZOX, called, respectively the XY-plane, YZ-plane and the ZX-plane,
are known as the three coordinate planes.
• The distances measured from XY-plane upwards in the direction of OZ are taken as positive and
those measured downwards in the direction of OZ′ are taken as negative.
• The distance measured to the right of ZX-plane along OY are taken as positive, to the left of ZX-
plane and along OY′ as negative, in front of the YZ-plane along OX as positive and to the back
of it along OX′ as negative.
• The point O is called the origin of the coordinate system.
• The three coordinate planes divide the space into eight parts known as octants. These octants
could be named as XOYZ, X′OYZ, X′OY′Z, XOY′Z, XOYZ′, X′OYZ′, X′OY′Z′ and XOY′Z′. and
denoted by I, II, III, ..., VIII , respectively




Coordinates of a Point in Space
• Choose a rectangular coordinate system.
• Let P be a point with coordinates by an ordered 3-tuple (x, y, z).