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I wrote a vector as $(a, b)$ in a maths exam but I was told it is not vector notation. However, I have seen this notation used all over this site and elsewhere. Is it correct or not?

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3 Answers

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This is a cultural thing. I think it's perfectly fine, in general. However, to give you some examples of why someone could find it non-fine:

1. In secondary school (high school) here in Norway, students learn that the coordinates of points are written $(a, b)$, and the coordinates of vectors are written $[a, b]$. As my calculus book says, this leads to pedantic double bookkeeping, as there is no mathematical reason to distinguish between points in the plane and vectors in the plane.

2. In linear algebra, the coordinates of your everyday vectors are usually written in column form: $\left(\begin{smallmatrix}a\\b\end{smallmatrix}\right)$, or $\left[\begin{smallmatrix}a\\b\end{smallmatrix}\right]$. Writing it as $(a, b)$ makes it a row vector, which is a different kind of object.

3. As pointed out in the comments: Some don't like using coordinates at all, and want you to explicitly write your vectors as linear combinations of the unit vectors, giving something along the lines of $a\vec i + b\vec j$ or $a\vec{e_x} + b\vec{e_y}$.

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Notation is (almost) never unilateral, there are good reasons for choosing one or another. Lastly, if you use a different notation to the agreed one, you should declare it.

For instance, consider the set $\mathbb{R}^2:=\mathbb{R}\times \mathbb{R}$. Therefore the points of $\mathbb{R}^2$ are, by definition of cartesian product, the couples $(a,b)$ where $a$ is in the first copy of $\mathbb{R}$, and $b$ in the second copy.

Now, $\mathbb{R}^2$ can be given the structure of vector space, so now you have $\mathbb{R}^2$ with a rule for summing points of $\mathbb{R}^2$ and a rule of multiplying points in $\mathbb{R}^2$ by scalars. An element $v$ of this vector space (a vector, by definition), is still a point of $\mathbb{R}^2$, so to emphasize that you could write $v=(a,b)$ for some $a,b$.

But then, since most of the times we do matrix-vector multiplications in linear algebra, a convenient way of writing vectors might be $v=\begin{bmatrix} a\\b \end{bmatrix}$.

The thing with notation is that it has to be declared in advance in order to be understood by others, and you choose one instead of the others depending on your goal, what you want to emphasize.

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I’ve also been told by my math teachers not to write vectors using the $(a, b)$ notation in order to avoid confusion, since that notation is often used to represent a single point on an $xy$-coordinate plane. Another answerer mentioned that it seems like something that varies by country/region, which makes sense—in the USA (the part that I’m from, at least) I was taught to write vectors as either $<a, b>$ or $a\vec x + b\vec y$.

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