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Concept of space, plane, line and point

Space: The space is the limitless three dimensional expanses where all matter exists. This is the region of near-vacuum surrounding all bodies in the universe. In geometry, we study geometric figures within certain sets of points called spaces. There are three kinds of spaces in geometry. First, the very simple space is a line, in which we can a represent a real number. It is called a one-dimensional space. Second, the space is a plane, in which we can represent a point, line, rectangular, circles etc. It is called a two-dimensional space. Third, the three-dimensional space, which is simply called the space, in which we can represent a solid having length, breadth and height or thickness. It is the universal set of points, lines, planes, cubes, spheres and so on.

Plane: A plane is a flat surface. It has length and breadth and no thickness. It is also regarded as an entity of two dimensions.

Line: A line has length only and no breadth or thickness. It is regarded as an entity of one dimension. There are two types of lines.

First, Straight line: A straight line is the shortest distance between two given points. It is also called a right line. There are various kinds of lines.

Second, Curved line: A curved line is nowhere straight. There are endless varieties of curved lines.

Point: A point has no length, breadth and thickness i.e. it has no magnitude. It has position only. It is regarded as an entity of zero dimensions. A point is always named with capital letters.

State includes five axioms.

(1)A state line may be drawn from any point to any other point.

(2)A finite straight line may be extended at each end in a straight line.

(3)A circle can always be drawn with any given center and any given radius.

(4)All right angles are equal one to another.

(5)If a straight line meets two other straight lines so that the two adjacent angles on one side of it are together less than two right angles, the other straight lines, when extended, will meet on that side of the first straight line.

State the five postulates of incidence.

As a concrete geometrical conception space is regarded as a set of points and straight line and plane are considered subsets of this universal set. There are five postulates about this conception. These postulates are called incidence postulate which are given below:

Postulate-1: The space is a set of all points and plane and straight line are subsets of this set.

Postulate-2:for two different points, there is one and only one straight line on which both the points lie.

Postulate-3:For three different points not lying in the same straight line, there is one and only one plane line on which the three points lie.

Postulate-4: A straight line passing through two different points on a plane lie on that plane.

Postulate-i ) Space contains more than one plane.

ii) More than one straight line lies in each plane.

iii) The real numbers can be related to points on each straight line such that every point of the line corresponds to a unique real number and every real   number corresponds to a unique point of the  line

State the distance postulate

The concept of distance in geometry is also an elementary conception. There are some postulates about it.

Postulate-1: Every pair of points (P,Q) determines a unique real number which is called the distance  between P and Q and is represented by PQ.

 

Postulate-2: If  P and Q  are different points, the number PQ is positive. Otherwise, PQ=0.

Postulate3: The distance between P and Q and that between Q and P are same,i.e. PQ= QP .

 

State the ruler postulate

One-to-one correspondence can be established between the set of points in a straight line and the set of real numbers such that, for any point P and Q,

PQ=  |P-Q|, where the one-to-one correspondence associates points P and Q to numbers p and q respectively. This postulate is known as ruler postulate.

 

Explain the number line

Number line: One-to-one correspondence can  be established between the set of points in a straight line and the set of real numbers such that for any point P and  Q ,PQ=|a-b|, where the one-to-one correspondence associates points P and Q to real numbers ‘a’ and ‘b’ respectively.

The correspondence mentioned in the proposition above is said to have reduced the line into a number line. If  p correspond to ‘a’ in the number line then p is called the graph point  of  P and P, the coordinates of ‘a’.

State the ruler placement postulate

Any straight line AB can be converted into a number line such that the co- ordinates of A are O and that of B is positive. This postulate is known as ruler placement postulate.

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