I recently answered a part of a question which required me to identify the "smallest" value from a set of real numbers (please forgive how simple that sounds) . I would like to know, does the word "smallest" imply the number with the least magnitude or does it include the least value (such as negatives).
For example is the smallest value from -3, 1, and 2 the value 1 or -3?
As a corollary, is it correct to choose a "smallest vector" and if so, do you then compare magnitudes?
Using the usual order on the integers or reals, the smallest of $\{1,2,-3\}$ is $-3$. I suspect many people would see it as magnitude and would answer $-1$. I would avoid the use of smallest when negative numbers are involved for this reason.
Similarly, for vectors it does not make sense to ask for the smallest. You should ask for the smallest magnitude, which is well defined. Many people will take the smallest vector to ask for the smallest magnitude.