Question

Show that $\arrowvert \vec a \arrowvert \vec \ b+ \arrowvert \vec b \arrowvert \vec \ a$ is perpendicular to $\arrowvert \vec a \arrowvert \vec \ b- \arrowvert \vec b \arrowvert \vec \ a$ , for any two nonzero vectors $\vec a \ and \ \vec b$.

Answer 1

If I want to use the dot notation for the time derivative of a vector is better (more common) to put the dot over the vector, or the other way around

\dot{\vec{v}}



\vec{\dot{v}}



The first says the rate of change of the vector components, and the second says a vector made from the component rates.

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askedFeb 9 '12 at 22:01



ja72
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closed as off topic by Matthew Leingang, Loop Space, diabonas, lockstep, Torbjørn T. Feb 11 '12 at 12:40

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7

This to me seems like a very good example of why it might be better to use bold for vector notation, as is the case with most textbooks, as opposed to arrows notation. – Peter Grill Feb 9 '12 at 22:39 

18

The derivative of a vector valued function does not depend on the choice of a basis. So the second notation means nothing: \vec{v} is the function that you're differentiating. – egreg Feb 9 '12 at 22:44 

2

Obviously, since you are taking a derivative of a vector and the result is a vector, you should use \vec{\dot{\vec{v}}}. – Jan Hlavacek Feb 10 '12 at 2:25

1

Using arrows instead of a bold font is like using underlines instead of italics -- useful in handwriting, but definitely not for printing. – Javier Bezos Feb 10 '12 at 9:31

1

@PeterGrill: Ok, I guess that's a good reason, but then how do you denote matrices of vectors? Or in case you have a matrix of physical values that are usually defined with capital letters? Also, from my experience students tend to overlook more easily the fact that a quantity is a vector when it's in bold. Especially so, if they copy a formula from a textbook and make a terrible mess of whether it's a vector or the scalar value of a vector! Therefore I think the arrow notation is much more (how shall I put it?) 'poignant'. – Count Zero Feb 10 '12 at 13:06

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I’ve only ever seen it written with the dot above the arrow; never below it. As @PeterGrill pointed out, this might be a good case to use boldface instead of arrows.

That said, I personally feel that arrow-vectors are under-used in TeX because the default implementation (\vec) looks awful. With some tweaks, it can be made to look better.

About TeX's \vec macro

One reason \vec looks bad is because the arrows are always the same width. Notice, for example, how \vec{w} and \vec{m} look:



The ‘m’ looks like it’s wearing a t-shirt three sizes too small, and the ‘w’ looks like it's having trouble balancing a plate on its head.

Let’s look at a more real-life example. Notice how the arrow sits too far to the right of the ‘r’ and ‘v’ below. Worse yet, the arrowheads are crooked, and the dots atop them are too high and are mis-aligned:



Alternative 1 — esvect's \vv macro

To improve upon this, the LaTeX2e package esvect provides 8 new variants of vector styles. Shown below is style b, which in my opinion looks the best of the bunch. You can write \vv{PQ} and the arrows are automatically drawn as wide as needed. However, note that the arrows are far too wide for vector variables, and also note that the horizontal bar of the arrow is composed of multiple overlapping line segments, resulting in visible anti-aliasing rendering artifacts when viewed on the screen:



I don’t recommend using \vv unless you really like the super-wide arrows. You might also want to check out the other 7 variants.

Alternative 2 — new macro \xvec

To address the shortcomings of \vecand \vv, here is a new command called \xvec (for lack of a better name) which measures its argument automatically and provides appropriate alignment for dots:



There are three forms:

\xvec{} makes an unadorned vector variable.

\xvec[.]{} makes a single-dotted vector variable.

\xvec[:]{} makes a double-dotted vector variable.



The sample file (included at the bottom of this reply) includes definitions to override TeX's default \vec (if you prefer) and also provides shortcuts \dvec and \ddvec for single- and double-dotted vectors, respectively.

Here are the 26 letters ‘a’ to ‘z’:



I’m not terribly happy with the ‘i’ and the ‘j’, but these are almost always typeset in boldface anyway—especially when used with a hat