2.2 Points and vectors
All points and vectors we work with are defined relative to some coordinate system.
The coordinate system has an origin and some axes emanating from such origin.
The axes are usually oriented at right angles to one another.
We usually think of a coordinate system as three axes emanating from an origin.
But in fact, a coordinate system is located somewhere in the world, and its axes are best described by three vectors that point in mutually perpendicular directions.
So, we extend the notion of a 2D/3D coordinate system to that of a 2D/3D coordinate frame.
For example, a 3D coordinate frame consists of a specific point "O", called the origin and three mutually perpendicular vectors, called a, b, and c (see figure 2.4).
It is convenient to use arrows to graphically represent a vector.
For example a vector v can be represented as: v=v1 a + v2 b+ v3 c, or v=(v1,v2,v3), where v1, v2 and v3 are the components in the underlying coordinate system, for instance v=(3,2,7) [1,7,5].
2.2 Points and vectors
Figure 2.4: A 3D Coordinate frame positioned in "the world"
A point has two important aspects.
First, a point suggests the notion of place or location.
This is its fundamental geometric aspect.
2.2 Points and vectors
Second, a point has talent quantitative properties; it is specified by a set of one or more real numbers: its coordinates. This is its analytical aspect.
For example a point p can be defined as: p=(x1,x2, ….,xn), where x1,x2, ….,xn are the real number coordinates of p and "n" is the number of dimensions of the coordinate system.
You can define absolute points or relative points.
This distinction arises because of the way point coordinates are computed and the way points are plotted or displayed.
Absolute points are given directly by their coordinates, for example in 2D, a set of absolute points can be given as: pi=(xi,yi), where i=1,….n. on the other hand, the coordinates of each relative point are defined relative to the coordinates of the point preceding it.
This is best demonstrated by the following expression:
 . Where pi is some initial point (origin),  . This is illustrated in figure 2.5.
|
Figure 2.5: Relative points.
|
2.2 Points and vectors
A vector on the other hand is a geometric object represented by an ordered set of numbers to which you assign certain properties.
The properties are direction and magnitude.
A vector is represented by a row or a column matrix, for example: A=[a b] is a 2D row matrix vector and B=[d e]T, where T means "transposition" is a 2D column matrix vector.
Each element of a row or column matrix is called a vector component.
Each vector component represents a displacement.
Imagine receiving a note with instruction to walk 30 paces to the east, then 40 paces to the north.
This is an example of a vector with a total displacement with 50 paces in a generally northeast direction.
The length of the arrow represents the length or magnitude of the vector and vector components, and the orientation of the arrow represents the direction (see figure 2.6).
Figure 2.6: Vectors and displacements
2.2 Points and vectors
A vector is often drawn as an arrow of a certain length pointing in a certain direction.
It is valuable to think of a vector geometrically as a displacement from one point to another.
For example, a vector V is the difference between two points p1 and p2 (see figure 2.7), where V = P2 - P1 = (x2 - x1, y2 - y1)= (Vx, Vy) and (Vx, Vy) are called the Cartesian components or elements which are calculated by subtracting the coordinates of the points individually.
Figure 2.7: A vector as a displacement
If a vector w is represented by w=(a,b,c) then the magnitude of w denoted by |w| and defined as the distance from the tail to the head of the vector is calculated by:
.
2.2 Points and vectors
For example the absolute magnitude of the vector say A=[8 3] is denoted by |A| and calculated by: |A|=
.
The absolute magnitude of another vector say B= [-8 -3] is calculated by |B|=
.
This means that both vectors A and B are parallel and have the same length (magnitude), but they are in opposite directions.