8.2 3D-Visualization of Vectors

The classical visualizing presentation of a vector is an array in space, whose length defines an absolute value and whose orientation defines a direction. The place at which the arrow is situated is arbitrary; one can for example let it start as zero-point vector from the origin of a Cartesian system of coordinates. Thus its endpoint (the tip of the arrow) is described by the three space coordinates x,y,z in this system of coordinates. Its length a, also referred to as the absolute value of the vector is obtained from the theorem of Pythagoras as a = x2 + y2 + z2

It obviously does not matter, how the system of coordinates, with respect to which the coordinates of the vector are defined , is orientated in space. Under a change of the coordinate system (translation of rotation), the individual coordinates also change, but position and length of the vector are affected by this. They are invariant under translation and rotation. This property provides the definition of a vector.

Quantities, that can be characterized via specifying a single number for every point in space, are called scalar in contract to vectors; an example would be a density- or temperature distribution.

The three-dimensional zero-point vector (3D zero-point vector) represents the position coordinates of a point in space. It is customary to write them as a matrix with only one column or line. As symbols on often uses a1,a2,a3 for the vector a or x11,x12,x13 for the vector xi. Thus the following representations are synonymous:

a = a1 a2 a3 = a1,a2,a3 , absolute value  a = a1 2 + a1 2 + a2 2 x1 = x11 x12 x13 = x11,x12,x13 , absolute value  x1 = x11 2 + x12 2 + x13 2

Symbols for the vector as a whole, like a and x1, were introduced at a time when they were written by hand. Some of the formats used back then, such as letters in cursive of with a arrow on top nowadays lead to a somewhat inconvenient typesetting situation, since they cannot be entered quickly on the the PC-keyboard. Thus we use, corresponding to the vector format of the formula editor MathType, bold letters in the font Times New Roman.

The absolute value of the vector (the length of the arrow) is symbolized by surrounding the vector by |-signs. This is analogous to the notation for the absolute value of complex numbers, but the notions of absolute value are not quite identical. The length of a vector is independent of its position relative to the origin of a coordinate system, while the absolute value of a complex number is always calculated from the origin. This difference falls away, if one writes a vector that starts from a point x1,y1,z1 and leads to a point x2,y2,z2 as difference of two zero-point vectors, i.e. x2 - x1,y2 - y1,z2 - z1.

The interactive 3D-simulation in Fig.8.1 trains the spatial perception of vectors. Pressing the button Random vector generates a zero-point vector with random integer coordinates (minimum -5,maximum 5) and represents it as a red arrow, embedded into a spatial tripod and supplemented by projections on the various coordinate planes, which can be switched on or off. It is advisable to pull this simulation to full screen size.

The coordinates of the vector are shown as projections onto the planes x = 0,y = 0 and z = 0 and are given in three coordinate fields. In these fields arbitrary different coordinates can be entered in order to study the effect on the position of the vector.

Alternatively the tip of the vector can be pulled with the mouse and the effect om the coordinates can be studied in two planes. The 3D-projection can be also be rotated in space with the mouse. In addition certain well-defined projections can be directly obtained via option switches.

Instructions for experiments can be found on the description pages of the simulation.


PIC
Figure 8.1: 3D-visualization of vectors in space: presentation of the components. The orientation of the projection can be adjusted with the mouse. The components of the red vectors can either be entered as numbers of created by a random number generator. The option boxes allow the selection of different projections.