Concepts Of Geometry



Concepts of geometry

Points, Lines & Planes  

The most fundamental geometric form is a point. It is represented as a dot with a capital alphabet which is its name. A line is a set of points and it extends in opposite directions up to infinity. It is represented by two points on the line and a double headed arrow or a single alphabet in the lower case. A plane is a two dimensional (flat) surface that extends in all directions up to infinity.

A plane has obviously no size and definitely no shape. However it is represented as a quadrangle and a single capital letter ( figure 1.1))

Figure  shows points A, D & Q, line AB, line l and plane P

Some axioms regarding points, lines and planes are given below.

• An infinite number of lines can be drawn through any given point.
• One and only one line can be drawn through two distinct points.
• When two lines intersect they do so at only one point.

Collinear And Coplanar

Three or more points are said to be collinear if a single line contains all of them. Otherwise they are said to be non collinear.

Figure  shows two lines l and m . Line l is such that it passes through A, B and C. Hence points A B and C are collinear. In the case of points P, Q and R there can be no single line containing all three of them hence they are called non-linear.

Similarly points and lines which lie in the same plane are called coplanar otherwise they are called non-coplanar.

Orders of magnitude

An order of magnitude is an exponential change of plus-or-minus 1 in the value of a quantity or unit. The term is generally used in conjunction with power-of-10 scientific notation.

In base 10, the most common numeration scheme worldwide, an increase of one order of magnitude is the same as multiplying a quantity by 10. An increase of two orders of magnitude is the equivalent of multiplying by 100, or 10². In general, an increase of n orders of magnitude is the equivalent of multiplying a quantity by 10ⁿ. Thus, 2315 is one order of magnitude larger than 231.5, which in turn is is one order of magnitude larger than 23.15.

As values get smaller, a decrease of one order of magnitude is the same as multiplying a quantity by 0.1. A decrease of two orders of magnitude is the equivalent of multiplying by 0.01, or 10^-2. In general, a decrease of n orders of magnitude is the equivalent of multiplying a quantity by 10^-n. Thus, 23.15 is one order of magnitude smaller than 231.5, which in turn is one order of magnitude smaller than 2315.

In the Standard International (SI) System of Units, most quantities can be expressed in multiple or fractional terms according to the order of magnitude. For example, attaching the prefix “kilo-” to a unit increases the size of the unit by three orders of magnitude, or one thousand (10³). Attaching the prefix “micro-” to a unit decreases the size of the unit by six orders of magnitude, the equivalent of multiplying it by one millionth (10^-6). Scientists and engineers have designated prefix multipliers from septillionths (10^-24) to septillions (10²⁴), a span of 48 orders of magnitude.