Final answer:
Key mathematical principles are at play here, including the commutative property of addition, multiplication of complex numbers, the concept of mutually exclusive events in probability, and the use of the Pythagorean theorem in vector addition.
Step-by-step explanation:
The question at hand involves understanding basic mathematical principles related to real numbers, addition of numbers, complex numbers, and the principle of mutually exclusive events. We are provided with several statements and must determine the truthfulness of each based on mathematical axioms.
Firstly, a and b are real numbers, and we are given several conditional statements about their relationship. We have two propositions to consider:
• The magnitude of force A must be greater than the magnitude of force B, which implies that a > b.
• The magnitude of force A must be equal to the magnitude of force B, suggesting that a = b.
However, without additional context or information about forces A and B, we cannot conclusively determine the relationship between a and b.
Regarding addition, the statement A+B = B+A is true due to the commutative property of addition, which states that the sum remains the same regardless of the order in which numbers are added.
For complex numbers, the equation A* A = (a + ib) (a - ib) = a² + b² is also true as it represents the multiplication of a complex number by its conjugate, resulting in a real number with the imaginary parts cancelling out. In probability, the concept of mutually exclusive events is exemplified by the statement that the probability of A and C occurring together is zero (P(A AND C) = 0) since they have no outcomes in common. Finally, the use of the Pythagorean theorem is true when determining the length of the resultant vector from the addition of two vectors that are at right angles to each other, as this is the direct application of the theorem in vector addition.