Why cannot we add a scalar to a vector of the same dimensions?

Question

I assume we can multiply a vector by a scalar, but why cannot we add them if they belong to the same dimensions.

Answer 1

I don't have enough reputation to post this as a comment, but borrowing from this thread: Is a vector a scalar

In mathematics, a scalar is just a 1D vector, so you can add. In physics, vectors are relative to a particular coordinate system or basis. If you need clarification, please post a comment and I'll update.

Answer 2

This question is old but I'll answer it for those who might be wondering if this is possible

You can't.

$1≠ [1, 0, 0]≠\begin{pmatrix}1 & 0 & 0\\\ 0 & 1 & 0 \\\ 0 & 0 & 1 \end{pmatrix} $

The main difference between the two boils down to the fact that a Scalar is a member of $R^1$ space, in which the word "Direction" has no meaning whatsoever. The only have a magnitude. On the other hand, a Vector is a member of $R^2$ space, and the notion of direction is crucial to define vectors. Thus, a vector has a direction as well as a magnitude.

A consequence of this fact is that scalars are invariant under coordinate transformation such as rotation and vectors are not, e.g. Draw a point on a page and rotate it. The point looks the same throughout the rotation. Now, draw an arrow and rotate it. The arrow will point in different directions depending on what angle you choose to rotate it.

One can use scalars to denote vectors, but one must keep in mind they're not the same thing. Unit vectors serve the same purpose in $R^2$ space that the number $1$ does in $R^1$ space, but that doesn't mean they are the same things. The belong to different families.

Answer 3

Who says you can't add a scalar to a vector?

Nobody is who. But also, nobody has defined what it means to add a vector to a scalar. It's undefined. That leaves you completely free to come up with your own definition of vector-scalar addition if you like. The trick is, getting anybody else to be interested in it.

You'd have a slim chance of getting mathematicians interested if your vector-scalar addition has some surprising properties.

You'd have a better chance if you can show how it can be used to simplify existing solutions to interesting problems.

It would be a whole lot easier for you to get the world's attention if you have a PhD in Mathematics from some prestigious university or, if you're a grad student, and you can get your advisor to add his name to your paper.

You'd have a prize-winning idea (maybe you could even get physicists interested) if you can show how your new technique can be used to solve problems that previously were unsolved.

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A lot of well known mathematical techniques probably started the same way: "Why can't we take the square root of -1?" "Why can't I draw more than one line through a point that does not intersect a given, other line?" etc.

On the other hand, there probably were a whole lot more "Why can't..." ideas that we've never heard of because they didn't amount to anything useful.