Given line is $\large\frac{x-5}{3}=\frac{y+4}{7}=\frac{z-6}{2}$

Hence the point $(x,y,z)\;is \;(5,-4,6)$ and the direction ration of the parallel vector is $(3,7,2)$

Hence $\overrightarrow a = 5\hat i - 4\hat j + 6\hat k$ and the parallel vector is $ \overrightarrow v = 3\hat i + 7\hat j + 2\hat k$

Hence the required equation is $ \overrightarrow r = (5\hat i - 4\hat j + 6\hat k )+ t (3\hat i + 7\hat j + 2\hat k) $

where $t$ is any real number.