As an undergraduate, I clearly remember learning and using "hat notation" to describe unit vectors. That is, if $\vec{v}$ is any vector (in 2 or 3 dimensions) then $\hat{v}$ denotes the unit vector in the direction $\vec{v}$, i.e. $$\hat{v} = \frac{\vec{v}}{|\vec{v}|}$$

The "hat" was also used for the standard unit vectors in the direction of the $x$-, $y$-, and $z$-coordinate axes: $$\hat{i}=\langle 1,0,0 \rangle,\quad \hat{j}=\langle 0,1,0\rangle, \quad \hat{k}=\langle 0,0,1\rangle$$

Now that I am teaching multivariable calculus for the first time, I see no use of this notation in our textbook (Stewart, 8th Ed.), and I am wondering if this notational convention is something I picked up from my undergraduate Physics coursework. (Just to avoid misunderstanding: Stewart uses boldface $\bf{i}$,$\bf{j}$,$\bf{k}$ for the unit vectors but does not use the hat accent.) So my questions:

Is "hat notation" for unit vectors (including the standard unit vectors, but not limited to them) widely used in teaching multivariable calculus, or is this something only physicists use? Are there math textbooks that use it?

Edited to add: I thought I would mention the main reason I'm interested in this notation: it makes certain formulas involving projection much simpler. For example, if $\vec{u}$ and $\vec{v}$ are any two vectors then $$\operatorname{proj}_{\vec{u}}\vec{v} = \vec{v} \cdot \hat{u}$$ $$\operatorname{comp}_{\vec{u}}\vec{v} = \left(\vec{v} \cdot \hat{u}\right) \hat{u} $$

(compare $\frac{\vec{v} \cdot \vec{u}}{|\vec{u}|}$ and $\frac{\vec{v} \cdot \vec{u}}{|\vec{u}|^2}\vec{u}$, respectively). Likewise the angle between any two vectors $\vec{u}$ and $\vec{v}$ is given by $\cos(\theta) = \hat{u} \cdot \hat{v}$.

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    $\begingroup$ As far as I can tell it's not "standard" in calculus text books I've seen (Stewert, Apex, Wittman, Strang) but it does get taught. I certainly use the notation for unit vectors when teaching and I know several of my colleague do. It may be more standard in linear algebra. $\endgroup$
    – Nate Bade
    Jan 5, 2018 at 4:12
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    $\begingroup$ It's really hard to write boldface i,j,k in homework and on tests. I tend to use $\hat{x},\hat{y}, \hat{z}$ in place of the quaternionic notation. That said, I haven't seen the beautiful notation $\vec{A} = A \hat{A}$ notation much used in math texts. You're probably correct about the physics influence, it is where I learned it. $\endgroup$ Jan 5, 2018 at 6:25
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    $\begingroup$ I don’t know how I picked it up if it’s not, but we briefly mention unit vectors in the precalc classes I teach, and I use it for the class (even though, indeed, it’s not in the book). Dotless $\hat{\imath}$ via \hat{\imath} for style points, of course :) $\endgroup$
    – pjs36
    Jan 5, 2018 at 7:52
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    $\begingroup$ Perhaps it is "common" in physics or engineering, not mathematics. $\endgroup$ Jan 5, 2018 at 10:48
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    $\begingroup$ I thought the "hat" worked like this: The standard unit vectors in the direction of the $x$-, $y$-, and $z$-coordinate axes would be called $\hat{x}, \hat{y},\hat{z}$ $\endgroup$ Jan 5, 2018 at 21:04

2 Answers 2


I wouldn't say it's common (none of the calculus books on my shelf use it). It's in some math books (e.g., this one, and this one and this one) but it's mostly in physics books.


Echoing other comments and @ncr's answer: this notational convention is not unknown, but is not common in mathematical contexts.

Perhaps more to the point, although relatively experienced math people might guess what is intended, they'd need to ask to proceed with confidence, so whatever time was thought to be saved by the convenient notation might be lost by having to answer questions about it.

Also, the very-very-common convention to use "hat" for Fourier transform should give the Fourier-transform use vast precedence over a relatively trivial use of giving the unit vector corresponding to a vector, in my opinion. (The fact that people typically talk about vectors for a while before learning about Fourier transforms doesn't alter my opinion. It's not good to have temporary notation that becomes wildly misleading soon after.)

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    $\begingroup$ Well, the context between vectors and Fourier transforms is usually quite different. For instance, physicists frequently use the hat also to denote operators in quantum mechanics. It's also worth noting that the hat to denote Fourier transforms is seldom used outside mathematics. In engineering it is common to use lower-case letters for time-dependent quantities and upper-case letter for transformed quantities. Sometimes, tilde or calligraphic letters are used in physics and engineering to denote Fourier transform. $\endgroup$ Jan 29, 2018 at 16:14
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    $\begingroup$ As I said once to my students in a class about sensors where we had to discuss many different quantities: "Guys, we have no way to not reuse notation; I don't like using same symbols for different quantities, but we have too many of them with established overlapping symbols, so pay attention at the context!" $\endgroup$ Jan 29, 2018 at 16:17
  • $\begingroup$ @MassimoOrtolano, I interpreted the question as asking about mathematics proper... Yes, context is always the wisest determiner, certainly, but in my experience there is rarely a need for a special notation for the corresponding unit vector. Yes, also in mathematics proper it is sometimes useful to denote Fourier (or other spectral transforms) as letters rather than diacritical marks, of course, but "hat"'s use is incessant. $\endgroup$ Jan 29, 2018 at 20:07

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