Answer
$S$ spans $R^3$.
Work Step by Step
Assume the combination
$$a(6,7,6)+b(3,2,-4)+c(1,-3,2)=(0,0,0).$$
We have the system
\begin{align*}
6a+3b+c&=0\\
7a+2b-3c&=0\\
6a-4b+2c&=0
\end{align*}
Since the determinant of the matrix is given by
$$\left| \begin{array} {cc} 6&3&1\\7&2&-3\\6&-4&2 \end{array} \right|=-184\neq 0$$
then the system has unique solution, that is, $$a=0, \quad b=0, \quad c=0.$$
Consequently, $S$ is linearly independent and since $R^3$ has the dimension $3$ then $S$ spans $R^3$.