Find the vector equation of the plane through the point $(3\hat{i}+4\hat{j}-\hat{k})$ and parallel to the plane $\overrightarrow{\mathrm{r}}·(2\hat{i}-3\hat{j}+5\hat{k})+5=0$.

Find the vector equation of the plane passing through the point $(a, b, c)$ and parallel to the plane $\overrightarrow{r}·(\hat{i}+\hat{j}+\hat{k})=2$.

Find the vector equation of the plane passing through the point $(1,1,1)$ and parallel to the plane $\overrightarrow{\mathrm{r}}·(2\hat{\mathrm{i}}-\hat{\mathrm{j}}+2\hat{\mathrm{k}})=5$.

Find the equation of the plane passing through the point $(1,4,-2)$ and parallel to the plane $2x–y+3z+7=0$.

Find the equations of the plane passing through the origin and parallel to the plane $2x-3y+7z+13=0$.

Find the equations of the plane passing through the point $( -1,0,7)$ and parallel to the plane $3x-5y+4z=11$.

Find the equations of planes parallel to the plane $x–2y+2z=3$ which are at a unit distance from the point $(1,2,3)$.

Find the distance between the planes $x+2y+3z+7=0$ and $2x+4y+6z+7=0$.

Find the equation of the plane through the line of intersection of the planes $x+y+z=6$ and $2x+2y+4z+5=0$, and passing through the point $(1,1,1)$.