If you square the vector $\vec v = 12.5 \, \text{m s}^{-1} \, \hat i$, what direction is $\vec v$? Similarly, if you had a quantity divided by a vector, $\frac{13 \, \text{kg}}{1.6 \, \text{m}, \, \hat i}$, what is the direction?
I came across this in physics, and I have been treating the unit vectors almost as units, because it works mathematically for me (except when there's two directions). I know this is incorrect, which is what inspired this question.
Also, is my notation for writing vectors correct?
Your notation for vectors would go fine with most. However, you cannot multiply or divide by vectors; you can only multiply or divide a vector by a scalar. In your case, there is no meaning to "the square of 12.5 metres per second east" or "13 kilograms divided by 1.6 metres east", but "twice 12.5 metres per second east" does have a meaning: 25 metres per second east. Since the operations you want to do are not defined, there is no associated direction.
Intuitively, one way to see that a system of multiplication and division by vectors to yield vectors doesn't exist is that there are infinitely many unit vectors. This post has more on why division by vectors isn't defined.
There is one exception to this no-go principle: complex numbers are sometimes represented as vectors and they do admit multiplication and division. But since you put units in your quantities, they aren't complex numbers…