Final answer:
Axiom 5 of vector spaces guarantees the existence of a negative vector which, when added to the original vector, yields the null vector. Vector subtraction equates to adding the inverse of a vector. If two vectors are of equal magnitude but opposite in direction, their resultant vector magnitude is 0.
Step-by-step explanation:
The question asks about Axiom 5 of a vector space, specifically the existence of a negative of a vector. In a vector space, for every vector ℝ, there must exist a vector −ℝ such that when they are added together, the result is the null vector or zero vector. This is similar to scalar subtraction, where the inverse of a number (its negative) is added to the original number to yield zero (5 + (−5) = 0, for instance).
Vector subtraction involves adding the negative of a vector. If vector A is subtracted from vector B, written as B - A, it is equivalent to B + (−A). The negative vector has the same magnitude but the opposite direction of the original vector. Vector magnitude remains positive even when multiplied by a negative scalar.
If two vectors are equal in magnitude and opposite in direction, such as vectors U and W mentioned in the GRASP CHECK question, the magnitude of their resultant vector would be 0 because they cancel each other out. For the case of the GRASP CHECK, given that vector V adds no component in the direction opposing U and W, the final magnitude of the resultant vector will still be 0.