Why is vector notation not used in the velocity formula (Galilean Transformations)? First of all, I'm not that good at physics. This question has to do with a physics course I'm taking at a maths school.
With that said, I am currently learning about the Galilean transformations and I'm a little confused about the vector notation in my textbook.
So the velocity of an object moving in relation to an inertial reference frame $O$ is denoted by
$u=\frac{\Delta x}{\Delta t}=\frac{x_2-x_1}{t_2-t_1}$
Now this object simultaneously moves in relation to our second inertial reference frame $O'$ which is moving away from $O$ with a velocity of $\vec{v}$. So the velocity of the object in relation to $O'$ is:
$u'=\frac{\Delta x'}{\Delta t'}=\frac{x_2'-x_1'}{t_2'-t_1'}=\frac{x_2-vt_2-(x_1-vt_1)}{t_2-t_1}$
The equation continues but my question is why isn't $vt$ written as $\vec{v}t$ in the velocity equation as well?
 A: Probably because you are considering a 1D system.
A: What you've written is not quite right. It should be the reference frame $O'$ is moving away from $O$ at velocity $v$ and not $vt$. The latter is the distance separating the two frames.
Also, you should state that the frames are parallel to each other, otherwise you would need a orthogonal transformation to make them parallel. It's normal in these situations that you are using the same units in both frames to measure time and distance.
Also, it's useful in these situations to use subscripts to indicate which frame you're measuring in. For example, the velocity of an object in frame  $O$ is better written as $u_O$.
Thus, if your object is moving with velocity $u_O$ in frame $O$ and frame $O$ is moving away from frame $O'$ at velocity $v_{O'}$ then $u_{O'}= v_{O'} + u_O$.
Now, the arrows on top of $x$ and $v$ show that you are using a vector and not a number.
That you are considering a frame suggests that you are considering motion in 3d space rather than 1d space where ordinary numbers would generally be used. So you ought to use the vector notation because its vectors you are dealing with.
Also, I should add that some people don't use the arrow notation because its just extra fuss and just let context decide which letters denote vectors and which scalars.
The important thing is to use notation consistently, especially in a single piece of work, and if you're doing a course to use the notation that is recommended by them.
