Alternative University

Analytic Geometry

The Number Line

Coordinate Geometry
“In order to locate points, lines, and surfaces their positions must first be referenced to some well-known position.  The Cartesian coordinate system, commonly used in mathematics and graphics, locates the positions of geometric forms in 2-D and 3-D space.”
— 
Krivoshapko & Ivanov, Encyclopedia of Analytical Surfaces

Analytic geometry (also called analytical geometry) is the study of geometric shapes with numerical coordinates. The coordinates could be the rectangular coordinates we have been using in previous lessons:

Figure 1: Two-dimensional (2D) rectangular coordinates for analytic geometry. The origin is (0,0).

Figure 2: Three-dimensional (3D) rectangular coordinates for analytic geometry.

In basic mathematics, for two dimensions (Figure 1 above) the horizontal axis is called the x-axis, and the vertical axis is called the y-axis. In multivariable calculus o these axes may be referred to as E1 and E2 (for Euclidean space). In structural analysis o these axes may be referred to as X1 and X2. In row-oriented data sampling reports, the terms ordinate and abscissa may be used (left handed). Any symbols to denote these axes are allowed. Usage of x and y for variables often does not refer to these coordinate axes.

Each of these two coordinate axes is a number line:

Figure 3: The number line.

The number line is a straight line of uniformly spaced ordered points, each point representing a real number, with the number represented by each point less than the number represented by any point to the right of that point on a horizontal number line (this referred to as ordering of the points):

Figure 4: For any two real numbers x and y, x is less than y if the point representing x on the horizontal number line is to the left of the point representing y. [Wikipedia]

The real numbers includes integers, fractions, etc. Following are where some commonly used real numbers are on the number line (√2=1.414…, e=2.71828…, π=3.14159…):

Figure 5: Commonly used real numbers on the number line.

Since this is a line (a one dimensional vector space), we can consider a vector to be a line segment on this line, with magnitude and direction. Magnitude is the length of the vector, and direction in this case is in the positive or negative direction.

This is different than a regular line segment of a line, which has magnitude but not direction. A line segment can be thought of as pointing in either direction. A vector however, has direction (not just magnitude).

On the traditional horizontal number line we are discussing, a vector points either in the positive direction (toward the right), or in the negative direction (toward the left).

In one dimension (on the number line), each vector has one component. For example, a vector (1), of length (magnitude) 1, extends one unit in the positive direction (towards the right), a vector of length −2 denoted (−2) extends two units in the negative direction (towards the left), etc.:

Figure 6: Two one-dimensional vectors on the number line.

Adding one-dimensional geometric vectors simply involves sliding them end-to-end on the number line:

Figure 7: One dimensional vector addition: (1) + (2) = (3). [Wikipedia]

In that example, vector (1) moves one unit in the positive direction, and vector (2) moves two more units in the positive direction, adding together to form the new vector (3).

Subtraction is simply addition, with changing the sign of the vector being subtracted, and moving that distance along the number line:

Figure 8:  (3) + (−2) = (1). [Wikipedia]

The distance from one point to another point on the number line is the absolute value of the difference of their positions:

Figure 9: Distance on the number line. [Wikipedia]

Absolute value, denoted by vertical bars, is the positive number of the value between the vertical bars. For example, the absolute value |5| is 5, and the absolute value |−5| is also 5.


Exercise

Problem: Calculate the distance between positions x = (−1.25) and y = (1.9) of Figure 9. Does the order of subtraction matter?

Solution: The order of subtraction does not matter, because an absolute value is taken of the result. We show both ways, first subtracting y from x:

distance  
 
 
 
 
|𝑥 − 𝑦|
|𝑥 + (−𝑦)|
=  |−1.25 + (−1.9)|
=  |−3.15|
=  3.15

Subtracting x from y produces the same result:

distance  
 
 
 
 
 
|𝑦 − 𝑥|
|𝑦 + (−𝑥)|
=  |1.9 + (−(−1.25))|
=  |1.9 + 1.25|
=  |3.15|
=  3.15

Scalar multiplication is as before, but in one dimension there is only one vector component:

λ( a )  =  ( λa )

Scaling by a negative number reverses the direction of the vector, while scaling the length by the absolute value of the scalar.

For example, the vector (1.5) points 1.5 units in the positive direction (toward the right in the number line). Multiplying that vector by −2 produces the vector (−3), which points 3 units in the negative direction (toward the left).

Scalar multiplication can be performed geometrically using similar triangles:

[+] Show multiplication using similar triangles

However, for this discussion, algebraic multiplication is sufficient. We will cover similar triangles later.


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2020–Oct–23  22:18  UTC