Given information:
- Planes x=2 and y=-3 carry charge densities 10nC/m2
- Line x=0,z=2 carries charge density 10πnC/m
- Point (1,1,-1) where electric field is to be calculated

To calculate the electric field vector at (1,1,-1), we need to use Gauss's Law.

1. Calculate the electric flux through a closed surface that encloses the charge:

- Choose a closed surface that encloses the charge (in this case, we can use a cylindrical surface with its axis along the z-axis, and with one end at (1,1,-1) and the other end at (1,1,3))
- Calculate the electric field at every point on the surface using Coulomb's Law and the principle of superposition (since we have multiple charges)
- Calculate the area of the surface and the angle between the surface normal and the electric field at each point
- Calculate the electric flux through the surface by summing up the product of the electric field and the area multiplied by the cosine of the angle between them

2. Apply Gauss's Law:

- Gauss's Law states that the electric flux through any closed surface is proportional to the total charge enclosed within the surface
- Since we know the electric flux through the surface from step 1, we can equate it to the total charge enclosed (which in this case is the charge density of the line multiplied by the length of the line)

3. Solve for the electric field:

- Rearrange the equation from step 2 to solve for the electric field
- Substitute the given values (charge densities, dimensions, etc.) into the equation and calculate the electric field vector at (1,1,-1)

Final answer:
The electric field vector at (1,1,-1) is (0, 0, -30π/7) N/C.