follows that the conditions $(3.1.1)- (3.1.3) $ are satisfied. By a direct computation we verify that the relations 3.1.4(1) - 3.1.4(4) from Definition 3.1 hold. Hence $V\times V$ is a vector groupoid called the [*coarse vector groupoid*]{} or [*pair vector groupoid*]{} associated to $V$.$\b$

[**The vector groupoid $V^2(p,q)$**]{}. Let $V$ be a vector space over a field $K$ and let $p,q\in K$ such that $pq=1$. The maps $\alpha,\beta,\iota:V^2\longrightarrow V^2$, $\alpha(x,y):=(x,px)$, $\beta(x,y):=(qy,y)$, $\iota(x,y):=(qy,px)$ together with the multiplication law given on $V^2_{(2)}:=\{((x,y),(qy,z))\ |\ x,y,z\in V\}\subset V^2\times
V^2$, by $(x,y)\cdot(qy,z):=(x,z)$ determine on $V^2$ a structure of vector groupoid. This is called the [*pair*]{} or the [*coarse vector groupoid of type $(p,q)$*]{} and it is denoted by $V^2(p,q)$.\
If $p=q=1$, then the vector groupoid $V^2(1,1)$ coincide with the pair vector groupoid associated to $V$(see Example 3.3 ).\
If $n$ is a prime number and $p,q\in \mathbb{Z}_n$, such that $pq=1$, then $\mathbb{Z}_n^2(p,q)$ is called the *modular* or *cryptographic vector groupoid*.$\b$

Let $V$ be vector space over a field $K$. One consider the maps $\alpha, \beta, \iota:V^3\longrightarrow V^3$, $\alpha(x_1,x_2,x_3):=(x_1,x_1,0)$, $\beta(x_1,x_2,x_3):=(x_2,x_2,0)$, $\iota(x_1,x_2,x_3):=(x_2,x_1,-x_3)$ together with the multiplication law given on $V_{(2)}^3=\{((x_1,x_2,x_3),(x_2,y_2,y_3))~|~ x_1,x_2,x_3,y_2,y_3\in V\}\subset V^3 \times V^3$ by $(x_1,x_2,x_3)\odot(x_2,y_2,y_3):=(x_1,y_2,x_3+y_3)$.

Then $(V^3,\alpha,\beta,\iota,\odot,V_0^3)$, where $V_0^3=\{(x,x,0)\ |\ x\in V\}$, is a vector 