Introduction to Three Dimensional Geometry, Objectives, Coordinate System and Coordinates of a Point in Space

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Objectives

After studying this lesson, you will be able to:

After studying this lesson

After Studying this Lesson

Coordinate System and Coordinates of a Point in Space

  • Recall the example of a bouncing ball in a room where one corner of the room was considered as the origin.

  • It is not necessary to take a particular corner of the room as the origin. We could have taken any corner of the room (for the matter any point of the room) as origin of reference, and relative to that the coordinates of the point change.

  • Thus, the origin can be taken arbitrarily at any point of the room.

The example of a bouncing ball in a room

The Example of a Bouncing Ball in a Room

  • Let us start with an arbitrary point in space and draw three mutually perpendicular lines and throug.

  • The point is called the origin of the co-ordinate system and the lines and are called the -axis, the -axis and the -axis respectively. The positive direction of the axes are indicated by arrows on thick lines in Fig.

The plane determined by the

  • -axis and the -axis is called -plane ( plane) and similarly, -plane (-plane) and -plane (-plane) can be determined.

  • These three planes are called co-ordinate planes. The three coordinate planes divide the whole space into eight parts called octants.

Co-ordinate planes

Co-Ordinate Planes

  • Let be any point is space. Through draw perpendicular on -plane meeting this plane at . Through draw a line parallel to cutting in .

  • If we write and , then are the co-ordinates of the point .

Again, if we complete a rectangular parallelepiped through with its three edges and meeting each other at and as its main diagonal then the lengths i.e., are called the co-ordinates of the point .

Rectangular parallelepiped OA,OB and OC

Rectangular Parallelepiped OA,OB and OC

Note: You may note that in Fig.

  1. The co-ordinate of the length of perpendicular from on the -plane.

  2. The co-ordinate of the length of perpendicular from on the -plane.

  3. The co-ordinate of the length of perpendicular from on the -plane.

  4. Thus, the co-ordinates and of any point are the perpendicular distances of from the three rectangular co-ordinate planes and respectively.

  5. Thus, given a point in space, to it corresponds a triplet called the co-ordinates of the point in space.

  6. Conversely, given any triplet, there corresponds a point in space whose co-ordinates are.

Example:

Name the octant wherein the given points lies: (a) (b)

Solution:

(a) Since all the co-ordinates are positive, lies in the octant.

(b) Since is negative and and are positive, lies in the octant.

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