NOTES : "n-dimensional vectors"

http://mathinsight.org/n_dimensional_vector_examples

Examples of n-dimensional vectors

At first, it may seem that going beyond three dimensions is an exercise in pointless mathematical abstraction. One might thing that if we want to describe something in our physical world, certainly three dimensions (or possibly four dimensions if we want to think about time) will be sufficient. 

It turns out that this assumption is far from the truth. To describe even the simplest objects, we will typically need more than three dimensions. In fact, in many applications of mathematics, it is challenging to develop mathematical models that can realistically describe a physical system and yet keep the number of dimensions from becoming incredibly large. The following simple examples reveal how quickly the required number of dimensions increases as we try to describe physical objects.

Position of a rigid object

How many dimensions does it take to specify the position of a rigid object (for example, an airplane) in space? Naively, one would think that it would take three dimensions: one each to specify the x-coordinate, y-coordinate, and the z-coordinate of the object. It is correct that one needs only three dimensions to specify, for example, the center of the object. However, even if the center of a rigid object is specified, the object could also rotate. In fact, it can rotate in three different directions, such as the roll, pitch, and yaw of an airplane. Consequently, we need six dimensions to specify the position of a rigid object: three to specify the location of the center of the object, and three to specify the direction in which the object is pointing.