Let $v,w$ be vectors of some vectorial space $V$. If $v=w$, are they said to be equivalent?

Question

Of course two geometrical vectors are called equivalent if they have the same magnitude, direction and orientation. But what about a generic vectorial space? Does the relation $v=w$ keep this name? I couldn't find the answer on my book or the internet.

Answer 1

Here's an example of an equivalence relationship $\sim$ on $V$ that isn't equality (although equality is an equivalence relation, just a boring one):

Let $W$ be a subspace of $V$. We define $v_1\sim v_2$ iff $v_2=v_1+w$ where $v_1,v_2\in V$ and $w\in W$. We then define the quotient vector space $V/\sim$.

As an example, $V=\mathbb{R}^2$ and $W$ is a line passing through the origin. Then the elements of $V/\sim$ are just the different lines in the plane parallel to $W$.