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Introduction to Three Dimensional Geometry

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Introduction to Three Dimensional Geometry - Lesson Summary

In three-dimensional space, we identify a point with respect to the distances from the three reference planes.

Take two planes perpendicular to each other.

The intersection of these two perpendicular planes is a straight line.

Let's take another plane perpendicular to both the existing planes.

These lines are called axes, and are named as X, Y and Z.

The point of their concurrence is called the origin.

The origin is denoted by the letter O. By convention, we consider the X—Y plane as the horizontal plane. Distances are measured starting from the origin.


Distances measured in one direction are taken as positive, and in the opposite side as negative.

For example, the distance measured from the origin along the X-axis along the direction shown is taken as positive.

The distance measured from the origin in the opposite side is considered negative.

To indicate this, we assign the symbol X'.

A similar convention is followed for calculating the distances along the Y and Z axes.

The plane formed by the X and Y axes is called the XY plane.

Similarly, the plane formed by the Y and Z axes is called the YZ plane, and the plane formed by the Z and X-axes is called the ZX plane.

The XY plane, YZ plane and ZX plane divide space into eight octants.

XOYZ is an octant. This octant is enclosed by the XYZ axes.

Similarly, the other octants are named according to the axes that enclose them. Any point in space is represented with the help of the distances that it measures with respective to the planes. These distances are called the coordinates of the point.

Suppose we have a point (x, y, z) in space. The coordinates of this point are x, y, z.
Each coordinate individually indicates the shortest distance of the point from the corresponding plane.

The coordinate, x, determines the shortest distance of the point from the Y—Z plane.

Similarly, the y co-ordinate represents the distance of the point from the Z—X plane, and z represents the distance of the point from the X—Y plane.

The coordinates of the origin are (0, 0, 0).

Using the coordinate axes in locating a point in the coordinate space:

First method is to drop a perpendicular on the X—Y plane. Let this perpendicular meet the X—Y plane at point Q.

Then we drop a perpendicular from Q on the X-axis. This meets at a point, say, R.
Distance OR is the x coordinate, RQ is the y coordinate and PQ is the z coordinate.

A point can be traced if the coordinates of a point are given.

First, identify a point, say R, on the X-axis, which is at distance x from the origin.
Next, we draw a perpendicular of length y from point R and parallel to the Y-axis on the X—Y plane to a point, say Q.

Lastly, we draw a perpendicular of length z from point R perpendicular to the X—Y plane.

Property among the coordinates of the points:

Any point that lies on the X-axis will have its coordinates in the form (x,0,0).
For example, a point at a distance of six units from the origin, on the x axis will have the coordinates (6,0,0)

Similarly, the points that lie on Y-axis will have coordinates of the form (0,y,0), and the points on the Z-axis will have coordinates of the form (0,0,z) Here, the letters, x, y and z belong to the set of real numbers.

Any point on the XY plane will be of the form, (x,y,0). The points (2, 2, 0), (6, -3, 0) lie on the XY plane.

Any point on the YZ plane will be of the form, (0,y,z) and a point on the ZX plane will be of the form (x,0,z). Here, the letters, x, y and z belong to the set of real numbers.

Locating a point using the intersection of planes:

First, take a plane at a distance x and parallel to the Y—Z plane.
Next, a plane parallel to the X—Z plane and at a distance of y is taken.

Finally, another plane parallel to the X—Y plane and at a distance of z is taken.

First, take two planes - one parallel to the Y—Z plane and the other parallel to the X—Z plane.

The intersection will be a line parallel the X-axis.

Likewise, the intersection of the other planes creates lines as shown.

The concurrence of these lines is point P.

The coordinate axes divide the coordinate space into eight parts, called octants. The sign of the coordinates determine the position of a point in a particular octant. For example, the point (2,-3, 3) lies in the fourth octant.

The x coordinate of the point is 2. So we measure 2 units along the X-axis and mark a point.

The y coordinate is -3. So, from this point, we measure (2, 3) on the X—Y plane in the direction of OY' parallel to the Y-axis. We mark a point there.

For the z coordinate, that is, 3, we measure 3 units from the point perpendicular to the X—Y plane.

We can notice that the point lies in the fourth octant.

The table shown gives the details of the octant according to the signs of the coordinates of a given point.

                \ Co-ordinates
Octants\        I        II      III      IV       V        VI      VII      VIII                 x    +     –     –    +    +     –     –    +                 y    +    +     –     -    +    +     –     –                 z    +    +    +    +     –     –     –     –

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