Introduction:
In this problem, we are given a small electric dipole placed in the xy plane at the origin with its dipole moment directed along the positive x-axis. We need to determine the direction of the electric field at a specific point (2, 2√2). To find the direction of the electric field, we will use the concept of superposition of fields due to the positive and negative charges of the dipole.

Given Information:
- Small electric dipole placed at the origin in the xy plane.
- Dipole moment directed along the positive x-axis.
- Point of interest: (2, 2√2).

Electric Field due to a Dipole:
The electric field at any point due to a dipole can be calculated using the equation:

E = (1/4πε₀) * ((p/r³) - (3(r⋅p)r/r⁵))

where:
- E is the electric field vector.
- ε₀ is the permittivity of free space.
- p is the dipole moment vector.
- r is the position vector from the dipole to the point of interest.

Calculating the Electric Field:
We can calculate the electric field at the point (2, 2√2) by substituting the values into the equation mentioned above.

1. Calculate the position vector:
The position vector r is the vector from the dipole (origin) to the point of interest (2, 2√2).

r = (2 - 0)i + (2√2 - 0)j
r = 2i + 2√2j

2. Calculate the dipole moment:
The dipole moment is directed along the positive x-axis, so the dipole moment vector p will be in the positive x-direction.

p = px = pxi

3. Substitute the values into the electric field equation:
E = (1/4πε₀) * ((px/r³) - (3(r⋅px)r/r⁵))

4. Simplify the equation:
Since the dipole moment is directed along the positive x-axis, the dot product r⋅px will be zero because the y-component of r is perpendicular to the x-axis component of px.

E = (1/4πε₀) * ((px/r³) - (3(r⋅px)r/r⁵))
E = (1/4πε₀) * (px/r³)

Result:
The electric field at the point (2, 2√2) is directed only along the positive x-axis due to the dipole.