Dot Product

A vector space over a field F is a set V together with two operations that satisfy the eight axioms listed below. Elements of V are commonly called vectors. Elements of F are commonly called scalars. The first operation, called vector addition or simply addition, takes any two vectors v and w and assigns to them a third vector which is commonly written as v + w, and called the sum of these two vectors. The second operation, called scalar multiplication takes any scalar a and any vector v and gives another vector av.

In this article, vectors are distinguished from scalars by boldface.[nb 1] In the two examples above, the field is the field of the real numbers and the set of the vectors consists of the planar arrows with fixed starting point and of pairs of real numbers, respectively.

To qualify as a vector space, the set V and the operations of addition and multiplication must adhere to a number of requirements called axioms.[1] In the list below, let u, v and w be arbitrary vectors in V, and a and b scalars in F.
Axiom Meaning
Associativity of addition u + (v + w) = (u + v) + w
Commutativity of addition u + v = v + u
Identity element of addition There exists an element 0 ? V, called the zero vector, such that v + 0 = v for all v ? V.
Inverse elements of addition For every v ? V, there exists an element ?v ? V, called the additive inverse of v, such that v + (?v) = 0.
Compatibility of scalar multiplication with field multiplication a(bv) = (ab)v [nb 2]
Identity element of scalar multiplication 1v = v, where 1 denotes the multiplicative identity in F.
Distributivity of scalar multiplication with respect to vector addition?? a(u + v) = au + av
Distributivity of scalar multiplication with respect to field addition (a + b)v = av + bv

These axioms generalize properties of the vectors introduced in the above examples. Indeed, the result of addition of two ordered pairs (as in the second example above) does not depend on the order of the summands:

(xv, yv) + (xw, yw) = (xw, yw) + (xv, yv).

Likewise, in the geometric example of vectors as arrows, v + w = w + v since the parallelogram defining the sum of the vectors is independent of the order of the vectors. All other axioms can be checked in a similar manner in both examples. Thus, by disregarding the concrete nature of the particular type of vectors, the definition incorporates these two and many more examples in one notion of vector space.

Subtraction of two vectors and division by a (non-zero) scalar can be defined as

v ? w = v + (?w),
v/a = (1/a)v.