Final answer:
Three vectors that sum up to zero satisfy the condition of closure, meaning they form a closed loop when positioned tail to head. The sum of zero indicates that vectors are balanced in such a way that, geometrically, they return to the starting point after following each vector sequentially.
Step-by-step explanation:
If three vectors sum up to zero, they satisfy the geometric condition of closure. This means that when the vectors are positioned tail to head, they form a closed shape or loop, without any gaps. Parallel vectors are those that point in the same direction, while orthogonal vectors are perpendicular to each other, forming a 90° angle. However, neither parallelism nor orthogonality alone guarantees that three vectors will sum to zero. A condition called antiparallel, where vectors point in opposite directions, combined with the right magnitudes, can lead to a sum of zero as well.
Furthermore, the commutative property of vector addition, which states that the order of addition does not affect the sum, also plays a role in understanding vector sums. Three vectors A, B, and C can be added in any order, such as A + B + C or B + C + A, and the resulting sum will be the same.
The case where three vectors sum up to zero illustrates a perfect example of vector equilibrium in two dimensions, where the resultant force or displacement is zero because the vectors exactly balance each other out.