How to distinguish 4D and 3D vectors in handwriting?

Usually vectors are denoted with bold font in printbooks and with arrows above in handwriting.

In Thorn's e al. Gravitation, 4D vectors are denoted with bold and 3D vectors with bold italic. How to implement similar distinguishing in handwriting?

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In component notation, 3d and 4d vectors are usually distinguished using latin and greek letters respecitively, e.g. $u_i$ and $u_\mu$.

Moreover, four-vectors without indices are usually just written as $u$, whereas three-vectors are denoted $\vec u$, as you say. You'll hardly find $\vec u$ denoting a four-vector.

The option $\underline{u}$ is also commonly used. Here you can also add more and more underlines for tensors - the number of underlines reflect the number of indices. $|u\rangle$ would also be possible, although it's mostly used in a quantum mechanical context.

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An additional option, particularly useful when you ALSO have 2D vectors (or higher dimensions) is to preface your vector's name with a subscript--e.g., ${}_{4}v$ and ${}_{3}v$, etc. It's also semi-common to use different symbols for your different metrics--$g_{ab}$ for the 4-metric, $\gamma_{ab}$ for the 3-metric and $q_{ab}$ for the 2-metric, for example. That way, you can distinguish the pullback of the 2-metric onto the 2-space from its representation in the 4-space through the index convention noted by Nick Kidman.

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I am using \x for fat x (and similarly for other letters) in handwriting, which works very well. I also use this in my theoretical physcis FAQ.

I also have LaTeX macros with the same abbreviations (except where backslash-letter already means something else, in which case I double the letter).

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