Is there a vector of magnitude $0$?

Question

(1) If I have a vector of the form $\vec v=a\hat\imath + b\hat\jmath$ and both $a$ and $b$ equal $0$ such that $\vec v=0$, is it still considered a vector?

(2) And, if so, does that make any value that is equal to $0$ a vector?

Answer 1

(1) Yes that it is still a vector, albeit a degenerate one.

(2) The vector $0 \hat{i} + 0 \hat{j}$ is technically a different object from the scalar/number $0$, even though they are often denoted the same (this is called abuse of notation). This is why I often prefer to write $\vec{0}$ for the vector $0 \hat{i} + 0 \hat{j}$, in order to differentiate from the number $0$.