Getting from vectors to a vector function

Given a set of velocity vectors and positions describing the non linear movement of an object in two dimensional space, is it possible to derive the vector function for the object?

For example:

Given the three following velocity vectors and corresponding positions,

$\displaystyle \vec{v_1}=\begin{pmatrix} 2 \\ 2 \end{pmatrix}$ from the point $\displaystyle p_1=(10,10)$

$\displaystyle \vec{v_2}=\begin{pmatrix} 2,82 \\ 0 \end{pmatrix}$ from the point $\displaystyle p_2=(12,12)$

$\displaystyle \vec{v_3}=\begin{pmatrix} 2 \\ -2 \end{pmatrix}$ from the point $\displaystyle p_3=(14,82,12)$

what is the vector function?

Thanks in advance.