Final answer:
The well-defined operations in the set of complex numbers C include addition, subtraction, multiplication, and division, with multiplication and modulus of vectors being exemplified. Complex number operations always result in another complex number.
Step-by-step explanation:
The operations that are well defined in the set of complex numbers C include addition, subtraction, multiplication, and division (except by zero). Every operation among complex numbers results in another complex number, which keeps the set closed under these operations.
For example, if A has real and complex parts (a + ib, where a and b are real constants), then A*A = (a + ib) (a - ib) = a² + b². This shows that multiplication of a complex number and its conjugate results in a real number, which is also a complex number where the imaginary part is zero. When performing operations like finding the modulus of a vector represented by complex numbers, components such as Cx, Cy, and Cz are involved, where the modulus of C is given by √((-2/3)² + (–4/3)² + (7/3)²) = √23/3.
With respect to vector operations, the addition of complex vectors follows the commutative property, such as A + B = B + A, and when multiplying by scalars, the distributive law applies, A(B+C) = A·B + A·C.