# 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 $\times$?

• It's fine; preferable, even. You won't use the "cross" in a class after High School. (But, you should use what your teacher wants in your class.) Commented Dec 14, 2014 at 16:11
• That's what I thought, but she keeps insisting that $\cdot$ is not correct, and that she will mark the answer as wrong on tests. Furthermore, she says that the same will happen in the exams I'll be making next year. Commented Dec 14, 2014 at 16:13
• If anything, the $\times$ should only be used with vectors. $\cdot$ is preferable, yup. The only thing where $\cdot$ can be confusing is in equations where scalars and tensors are mixed: There it is good practice to don't write anything for scalar multiplication, use $\cdot$ only for matrix multiplication or the scalar product of vectors and $\times$ never ever for anything other than the 3D vector cross product. Your teacher sounds inexperienced in exact sciences, but such things are hard to tell from hear-say. Commented Dec 14, 2014 at 16:14
• Chances are, your teacher prefers $\times$ because it is difficult to distinguish $3\cdot 4$ and $3.4$ from someone with poor handwriting. Commented Dec 14, 2014 at 16:14
• Fair, and that would apply to me; we use commas (3,4) in Portugal though. Commented Dec 14, 2014 at 16:16

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.

• That's a good point, but not what I meant... I mean that you can use $\cdot$ instead of $\times$. Commented Dec 14, 2014 at 16:20
• @someonewithpc I will add some comments about vector notation, too. Commented Dec 14, 2014 at 16:21
• Good answer. Let's not forget that $\times$ is also used to denote Cartesian products on sets---in fact, you use that interpretation above in $V\times V$! It is incorrect to use $V\dot V$ in that situation. So I agree with your larger point: parties need to agree on notation. Sometimes conventional wisdom will dictate notation; other times, it will be the person in authority. Commented Dec 14, 2014 at 17:58