What is the usual meaning of having the symbol $\hat{}$ (i.e., a hat) over a vector name? What do vectors denoted by $\hat{\mathbf{u}}$ usually represent?
For example, in this video, at min 3:00, the author denotes a unit vector by $\hat{\mathbf{u}}$, but I don't understand why not simply denoting it by $\mathbf{u}$. What is the difference between $\mathbf{u}$ and $\hat{\mathbf{u}}$?
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$\begingroup$In the video, at that very time, the speaker explains he is using that notation to point out $\hat{u}$ is a unit vector, i.e. a vector of norm $1$.
This sort of notations (once explained) helps remembering what are the specific properties of the quantities used in a proof. (Just looking at $\hat{u}$, you know it is unit.)
In linear algebra, the $\hat{}$ is widely used for unit vectors, so this is a general convention, at least in a good part of the world.
$\endgroup$ 2 $\begingroup$When $u$ is a mathematical object expressions like $\bar u$, $\hat u$, $\breve u$, $u'$, etc. usually denote new objects derived from $u$, or related to $u$, in some way, e.g., $$\breve f(x):=f(-x)\qquad(x\in{\mathbb R})\ ,$$ whereby the exact meaning is not an ISO standard, but is explained in the context. The overbar can denote the complex conjugate, in other circumstances a mean value, or new coordinate functions $(\bar x_1,\ldots,\bar x_n)$ replacing the present $(x_1,\ldots,x_n)$.
Contrasting this, the notation $\vec{u}$ just tells the reader that the object $u$ is a vector. In a time where the number of usable fonts is unlimited one may as well write ${\bf u}$ right from the start, if one wishes to make the distinction between scalars and vectors visible at first glance. But $u=(u_1,\ldots,u_n)$ is perfectly okay.
Given all this, the notation $\hat a$ leads to the following interpretations: It can mean (i) "I'm now introducing the vector $\hat a$, assumed to be a unit vector", or (ii) "Given any vector $a\ne0$ the vector $\hat a$ is defined by $$\hat a:={a\over\|a\|}\ .{\rm"}$$
$\endgroup$ 2 $\begingroup$If $\pmb u\in\Bbb R^n$ is any nonzero vector, then$$\hat{\pmb u}:=\frac{\pmb u}{\|\pmb u\|},$$where $\|\cdot\|$ denotes the euclidean norm: $$\|\pmb u\|=\|(u_1,...,u_n)\|:=(u_1^2+\cdots+u_n^2)^\frac12.$$
As an example, take $n=3$ and $\pmb u=(4,0,3).$ Then $\|\pmb u\|=\sqrt{4^2+0^2+3^2}$ and $\hat{\pmb u}=(0.8,0,0.6).$
$\endgroup$ $\begingroup$In the video you linked $\hat{u}$ represents the unit vector. A lot of times you will see the unit vectors in the $x,y,z$ direction written like this.$$\hat{i}=\begin{pmatrix}1\\0\\0\end{pmatrix},\hat{j}=\begin{pmatrix}0\\1\\0\end{pmatrix},\hat{k}=\begin{pmatrix}0\\0\\1\end{pmatrix}$$
Notice that this is just a convention. Here in Germany the unit vectors are written like this:
$$e_x, e_y,e_z\space \space \text{or} \space \space e_1,e_2,e_3 \space \space \text{or} \space \space \hat{e}_1,\hat{e}_2,\hat{e}_3$$
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