Final answer:
The Distributive Property is illustrated by the expression A(B + C) = AB + AC, while A+B = B+A shows the Commutative Property of Addition; both properties are fundamental in algebra and vector addition.
Step-by-step explanation:
To determine which property or operation completes the table, we must look at the equations and expressions provided in the question context and the patterns they exhibit. The expression A (B + C) = AB + AC is an example of the Distributive Property of multiplication over addition, which dictates that multiplying a number by a sum of numbers is the same as multiplying each addend individually and then summing the products. Similarly, the expression A+B = B+A illustrates the Commutative Property of Addition, where the order of terms can be swapped without affecting the outcome of the addition.
These examples demonstrate how certain properties, such as the distributive and commutative properties, are fundamental in the operations of addition and multiplication in algebra and vector operations. Specifically, the commutative property ensures that the order in which two vectors are added does not impact the resultant vector—a principle that is crucial in physics and engineering.