Properties of the non-commutative finite associative algebras of 3-dimensional vectors are presented. An interesting feature of these algebras is mutual associativity of all modifications of the defined parameterized multiplication operation and existence of a large set of single-side unit elements. In the ordinary case one unique two-side unit element is connected with every element of the algebra, except the elements that are square roots of zero element. It is shown that the used method suites for defining finite non-commutative associative algebras of arbitrary dimension m > 2: The considered finite associative algebras are interesting for cryptographic applications.