Unizor - Vectors - Congruence
Channel: Zor Shekhtman on UNIZOR Education
Date: 2013-12-12

After we introduced two major representations of vectors (geometric as arrows and analytic as tuples) we need to address an issue of congruence between vectors.
Obviously, if we speak about congruent vectors, we have to consider only vectors in the same space. We cannot consider a vector on a two-dimensional plane and a vector in a three-dimensional space as congruent because a concept of direction is completely different in these two spaces. Even if the dimensionality is the same, the qualitatively different spaces have different concepts of direction. For instance, a vector on a two-dimensional plane with Euclidian coordinates signifying the length cannot be congruent to a vector on another two-dimensional plane with Euclidian coordinates signifying a temperature and pressure of a gas in some reservoir.

When two vectors exist in the same space, the obvious definition of congruence is equality in two major vector characteristics - magnitude and direction. If vectors are represented analytically using tuples, congruent vectors must have the same real number on each place in the tuples that represent them. For geometrically represented vectors we have to resort to the same process of superposition we used when defined congruent geometric objects, like triangles. There is, however, a very important difference. In geometry we allowed different transformations that can be used to superimpose one object on the other - parallel shift, rotation and reflection. For vectors, where direction plays a very important role, we have to use only transformations that preserve the direction. The only transformation of this sort is a parallel shift. So, two vectors represented as arrows in the same space are congruent if and only if they can be superimposed after some parallel shift of one of them. It's easy to see that two congruent vectors form a parallelogram if we connect their origins and their arrow-pointed ends.

Let's go back to geometrical representation of a vector in the Euclidean space as a segment with an arrow at the end. Based on the principle of congruence, it makes sense to consider only vectors with an origin at the beginning of coordinates. That makes it easier to deal with both geometric representation of a vector as a directed segment and its analytic representation as a tuple, and it does not require to prove congruence by some transformation using parallel shift. Later on we will introduce operations on vectors and the only geometrical illustration we will use is the one with directed segments originated at point (0,0).