Answer
No.
Work Step by Step
We know that the determinant of a matrix \[
\left[\begin{array}{rr}
a & b \\
c &d \\
\end{array} \right]
\]
is $D=ad-bc$.
We know that if we choose $2$ of these vectors and the determinant of the matrix formed by the vectors is not $0$, then they span the plane. If we choose $(6,-2),(-2,2/3)$, then $D=6\cdot2/3-(-2)(-2)=4-4=0$, and $(6,-2)$ is parallel to $(3,-1)$. Thus it doesn't span it.
