Why do some people represent vectors with overbars while others use underlines?

Question

When I was originally introduced to vectors, I was told to write them with an arrow above the variable, like so: $$\vec{x}$$

As soon as I began taking vector-heavy classes, I found that those teachers dropped the arrow head: $$\overline{x}$$

Eventually, I also had a few professors who used underlines to represent vectors: $$\underline{x}$$

Around the same time as I started seeing the underline notation, I learned that these lines are all a substitute for bold letters, which work well in print, but not in handwriting: $$\bf{x}$$

While the rationale for dropping the arrow head seems apparent to me (less effort, doesn't crowd the area near the superscript), the rationale for preferring the overbar or underline is not. I use the underline notation myself, and while I have several reasons for preferring it, I seem to be in the minority. Is there a reason why both notation systems are in use at the same time?

Answer 1

According to Cajori's History of Mathematical Notations, v.2, §502 (1929) the overbar, overarrow and boldface notations are older, which may be one reason why they are more common. Shaw's 1912 survey of vector notations also mentions boldface, italics, Gothics, overbar, etc., but not underbar. Judging by How to represent a tensor/matrix/vector/array in blackboard? thread on Academia SE, the underbar is interpreted as a handwritten equivalent of boldface, and preferred by engineers. From the accepted answer by Massimo Ortolano:

"I usually employ underbarred symbols for four reasons:

1. Of course, I'm an engineer!
2. A bar is faster to draw then an arrow.
3. My handwriting is awful, and I think that simpler symbols improve readability; at the same time, though, I don't want to abandon the categorization of quantities through different symbols.
4. The interpretation of underbar is frequently that of bold, and I prefer bold symbols for vectors over symbols with an arrow."

Such field specific variations are not uncommon, compare to $i$ in mathematics vs $j$ in engineering for the imaginary unit. In more advanced mathematical texts and classes vector distinguishing marks are dropped altogether, and whether something is a vector is expected to be inferred from context.

Wikipedia describes some vector notation standardization efforts at the beginning of 20th century involving Macfarlane, Klein and Sommerfeld. Here is from Cajori:

"Argand adopted for a directed line or vector the notation $\overline{AB}$. Möbius used the designation $AB$, where $A$ is the origin and $B$ the terminal point of the vector. This notation has met with wide adoption. It was used in Italy by G. Bellavitis, in France by J. Tannery, by E. Rouche and Ch. de Comberousse and P. Appell. On the other hand, W. R. Hamilton, P. G. Tait, and J. W. Gibbs designated vectors by small Greek letters... We have already called attention to the designation $a\tiny{\alpha}$ of Frarncais and Cauchy, which is found in some elementary books of the present century; Frank Castle lets $A\scriptsize{\alpha}$ designate a vector. Frequently an arrow is used in marking a vector, thus $\vec{R}$, where $R$ represents the length of the vector.

W. Voigt in 1896 distinguished between "polar vectors" and axial vectors" (representing an axis of rotation). P. Langevin distinguishes between them by writing polar vectors $\vec{a}$, axial vectors $\overset{\scriptsize\circlearrowleft}{a}$. Some authors, in Europe and America, represent a vector by bold-faced type $\mathbf{a}$ or $\mathbf{b}$ which is satisfactory for print, but inconvenient for writing. Cauchy in 1853 wrote a radius vector $\overline{r}$ and its projections upon the co-ordinate axes $\overline{x}, \overline{y}, \overline{z}$, so that $\overline{r}=\overline{x}+\overline{y}+\overline{z}$. Schell in 1879 wrote $[AB]$; some others preferred $(AB)$."