1.1 Preliminary Remarks on Vectors 5
Furthermore, for any real numberkand two vectorsaandbone has:
k(a+b)=ka+kb. (1.6)Mathematical objects which obey the rules or axioms (1.1)–(1.6) are elements of a
vector space. For mathematician, the answer to the question “what is a vector?” is:
“it is an element of a vector space”. In addition to the arrows we discussed, there are
many other types of vector spaces. Examples are,
- real numbers or complex numbers,
- polynomials of ordern,
- quadratic matrices,
- ordered n-tuples(a 1 ,a 2 ,...,an)with real numbersa 1 ,a 2 toan.
In physics, the notionvectoris used in a more special sense. Before this is discussed,
a brief remark on thenorm of a vectoris in order.
1.1.2 Norm and Distance
It is obvious that an arrow has a length. For an elementaof an abstract vector space
thenorm||a|| ≥0 corresponding to the length of a vector has to be defined by rules.
Computation of the norm requires ametric. Without going into details, the general
properties of a norm are listed here.
- When the norm of a vector equals zero, the vector must be thezero-vector.Like-
wise, the norm of the zero-vector is equal to zero, thus,
||a|| = 0 ↔a= 0. (1.7)- For any real numberrwith the absolute magnitude|r|, one has:
||ra|| = |r|||a||. (1.8)- The norm of the sum of two vectorsaandbcannot be larger than the sum of the
norm of the two vectors:
||a+b||≤||a|| + ||b||. (1.9)
The relation (1.9) is obvious for the addition of he arrows as shown in Fig.1.1.
The distanced(a,b)between two vectorsaandbis defined as the norm of the
difference vectora−b:
d(a,b):= ||a−b||. (1.10)
For vectors represented as arrows with their tails located at the same point, this
corresponds to the length of the vector joining the arrowheads.