Electric Field and Equipotential Lines:
Electric field is a vector field that describes the magnitude and direction of the force experienced by a charge placed at a particular point in space. Equipotential lines are imaginary lines along which the potential difference between any two points is zero.

Electric Field Strength Vector:
The electric field strength vector at any point is defined as the force experienced by a positive test charge placed at that point divided by the magnitude of the test charge.

Calculating the Electric Field Strength Vector:
To calculate the electric field strength vector at point (1,2) in the given electric field, we need to determine the direction and magnitude of the electric field at that point.

Determining the Direction:
The direction of the electric field at any point is perpendicular to the equipotential lines passing through that point. Therefore, the direction of the electric field at point (1,2) is perpendicular to the line y=2x passing through that point.

Determining the Magnitude:
The magnitude of the electric field at point (1,2) can be determined using the formula E = -dV/dx, where E is the magnitude of the electric field, V is the electric potential, and x is the distance along the x-axis.

Since the equation of the equipotential line passing through point (1,2) is y=2x, we can write the electric potential at point (1,2) as V = k(2x), where k is a constant.

Differentiating this expression with respect to x, we get dV/dx = 2k. Substituting this value in the formula for the electric field magnitude, we get E = -2k.

Final Answer:
Therefore, the electric field strength vector at point (1,2) in the given electric field is -2k in a direction perpendicular to the line y=2x passing through that point.