Final answer:
The steepest tangent vector is not necessarily unique as functions can have multiple points with identical gradients. The uniqueness depends on the behavior and characteristics of the function itself.
Step-by-step explanation:
The question of whether the steepest tangent vector is unique in calculus can be answered with option (b) No. In many cases, a function can have multiple points where the gradient (which represents the direction and magnitude of the steepest climb) is the same. This means that there can be multiple steepest tangent vectors at different points of the function. However, this doesn't apply to all functions. For some functions, particularly those with unique maxima or minima the steepest tangent vector may indeed be unique.
The issue of uniqueness largely depends on the topology and shape of the function under consideration. For instance on a perfectly spherical hill, there are infinitely many steepest tangent vectors all pointing away from the center of the hill. This exemplifies that the uniqueness is not guaranteed.