Is it ever correct to say that $\vec{a}-\vec{a}=0$?

Question

My textbooks define $$\begin{cases}0\cdot \vec{a}=\vec{0}\\(m+n)\vec{a}=m\vec{a}+n\vec{a}\end{cases}$$ Therefore, $\vec{a}-\vec{a}=(1-1)\vec{a}=0\cdot\vec{a}=\vec{0}$.

But is it ever acceptable, in some context, to claim that $\vec{a}-\vec{a}=0$?

Answer 1

If I understand the question, you're asking whether it's acceptable to write $0$ rather than $\vec 0$. Typically, no: the first is a scalar, while the second is a vector, and the addition of two vectors gives another vector.

There are other common notations for vectors, such as bold type (e.g, $\mathbf a - \mathbf a = \mathbf 0$ for all vectors $\mathbf a$), but one way or another you must make the distinction between scalar and vector.

Answer 2

$0$ usually denotes the neutral element whenever you are working in a (additive) group, like in this case. It's just a short name for that (the zero vector $\vec{0}$) so as long as the context is clear, it's ok to use it.