Working with group theory I've found multiple times the idea of quotient group as $G/H = \{gH\ |\ g\in G, H < G\}$. Nevertheless, you can find similar things in vectorial spaces as $\mathbb{R}^2/L = \{P\vec{v} = \vec{v}^{\perp}\ |\ \vec{v} = \vec{v}^L + \vec{v}^{\perp},\ \vec{v}\in \mathbb{R}^2,\ \vec{v}^L \in L, \vec{v}^{\perp} \in L^{\perp}\}$. Here, $L$ is an straight line in $\mathbb{R}^2$
Is there a way to understand any quotient of this type without given the proper definition of the quotient (as I did above) nor equivalent relation? A general way to deduce what the quotient means by the meaning of the numerator and denominator?