Final answer:
Ordered triples of set A form elements of AxAxA, such as (a, b, c). The Cartesian product AxA corresponds to a 2-dimensional space with cardinality 16. Points are placed in quadrants based on their x and y coordinate signs, the x component is taken from a point in 3D space, and calculations of powers of i and complex numbers' magnitudes and components are performed following their respective mathematical rules.
Step-by-step explanation:
To understand the concept of Cartesian product and dimensions first, let's tackle the question what is an element of AxAxA? Since set A = {a, b, c, d}, AxAxA would represent the set of all possible ordered triples where each element is from set A. An example element of AxAxA would therefore be (a, b, c).
Concerning the question about the dimension of A, the set A does not have a dimension in the geometric sense as it's simply a set of distinct elements. However, when talking about the dimension of AxA, this refers to the Cartesian product of A with itself, which is a set of ordered pairs, and can be associated with a 2-dimensional space. Each element in AxA is a pair (x, y) where x and y are elements of A.
The cardinality of AxA is the number of elements in this set, which is the cardinality of A squared, hence with 4 elements in A, AxA would have 4^2 = 16 elements.
In the context of a Cartesian plane divided into four quadrants, the point (-4.5, 4.5) would be located in the second quadrant since the x-coordinate is negative and the y-coordinate is positive. The point (-1.2, -7.7) would be located in the third quadrant as both coordinates are negative. Conversely, the point (8.2, 9.1) would be located in the first quadrant since both coordinates are positive.
Regarding a point represented in classical Cartesian Space form like (-2.3, 3, -2.4), the x component of this point is -2.3.
The value of i^6, where 'i' is the imaginary unit (i^2 = -1), can be determined by realizing that i^4 = 1, so i^6 = i^2 * i^4 = -1 * 1 = -1.
For a complex number z= -π - 3.5i, the real component is -π, and the imaginary component is -3.5. The magnitude of a complex number z = -6 - 8i, denoted as |z|, is the square root of the sum of the squares of the real and imaginary parts, which is √((-6)^2 + (-8)^2) = √(36 + 64) = √100 = 10.
Lastly, given z = -2.5 - 3.2i, the value of z is simply the complex number itself, -2.5 - 3.2i. For z = 3 - 5i, the value of z + z is 2z, which would be 6 - 10i.