# Is “hat notation” for unit vectors commonly used in mathematics?

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 $\cos(\theta) = \hat{u} \cdot \hat{v}$.

• 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. – Nate Bade Jan 5 '18 at 4:12
• 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. – James S. Cook Jan 5 '18 at 6:25
• 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 :) – pjs36 Jan 5 '18 at 7:52
• Perhaps it is "common" in physics or engineering, not mathematics. – Gerald Edgar Jan 5 '18 at 10:48
• 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 $\hat{x}, \hat{y},\hat{z}$ – Gerald Edgar Jan 5 '18 at 21:04