Is there any reason I can't use the $\cdot$ (dot product sign) instead of $\times$ (multiplication sign)? Do note that I have read this question. However, I don't think it's quite the same question.
When dealing with simple number multiplication, I actively try to use $\cdot$ instead of $\times$. Take the following example:
3 $\times$ 4
as opposed to:
3 $\cdot$ 4
They give the same result. However it's easier to use $\cdot$, so I don't get x and $\times$ confused. When I asked my high school teacher, she said that I couldn't do this; to which I said: Why? She said that $\cdot$ should only be used for vectors, but isn't 3 just a 1D vector?
Should I stick with using $\cdot$ or switch to $\times$?
 A: The comments have addressed the notational issues, but I will discuss a comment you made.

Isn't $3$ just a 1D vector?

No.
$3$ is a scalar unless we define a 1-D vector space over the reals. There are some issues with considering every scalar a de facto 1D vector.
For instance, scalar-matrix multiplication is well defined, e.g. $3\begin{pmatrix} 1 & 0 \\ 0 & 1\end{pmatrix} = \begin{pmatrix} 3 & 0 \\ 0 & 3\end{pmatrix}$. If $3$ were a vector, we'd have a $1\times 1$ vector multiplying a $2\times 2$ matrix. This does not suit our definition of matrix-vector multiplication.

Another issue is the use of $\cdot$ and $\times$ in vector algebra. Strictly speaking, vectors do not get multiplied together. Vectors can be added together, and they can also be multiplied by scalars, viz.
$$\mathbf{u}+\mathbf{v} = \begin{pmatrix} u_1 + v_1 \\ u_2 + v_2 \\ \vdots \\ u_n+v_n \end{pmatrix}, \\
\alpha \mathbf{v} = \begin{pmatrix} \alpha v_1 \\ \alpha v_2 \\ \vdots \\ \alpha v_n \end{pmatrix}.$$
In vector algebra, $\cdot$ denotes the dot product or inner product between vectors, and if the vectors are 3-D or 7-D, $\times$ denotes the cross product. These are not multiplications in the conventional sense.
Namely, $(\cdot) : V \times V \to \mathbb{F}$, that is, the dot product maps two vectors from a vector space $V$ to a scalar in its underlying field $\mathbb{F}$, and $(\times) : V \times V \to V$, so the cross product maps two vectors in $V$ to another vector in $V$. These operations don't have inverses per se... there is no "dot division" or "cross division."
The familiar notation gets overloaded as we encounter more complex structures. Therefore, it is important to understand the context of what you're writing/reading.
For this reason, the teacher is not wrong to insist on a common notational style.
