Introduction:
In this scenario, a ray of light is incident along the y-axis on a medium with a variable refractive index. The refractive index of the medium changes according to the relation u = 1 x^2, where x is the x-coordinate. We need to determine the light ray vector at the point where the x-coordinate becomes equal to 1.

Understanding the Problem:
To solve this problem, we need to understand the concept of refraction and how the refractive index affects the direction of light rays. The refractive index is a measure of how much a medium can bend or refract light. When light travels from one medium to another with a different refractive index, it changes its direction.

Applying Snell's Law:
To determine the light ray vector at the given point, we can use Snell's law, which relates the angles of incidence and refraction to the refractive indices of the two media.

Step 1: Calculating the Angle of Incidence:
Since the ray of light is incident along the y-axis, the angle of incidence is 90 degrees.

Step 2: Calculating the Refractive Index at x = 1:
Given the relation u = 1 x^2, we can substitute x = 1 into the equation to find the refractive index at that point.
u = 1 x^2
u = 1 (1)^2
u = 1

Step 3: Applying Snell's Law:
Snell's law states that the ratio of the sine of the angle of incidence to the sine of the angle of refraction is equal to the ratio of the refractive indices of the two media.
sin(i) / sin(r) = n1 / n2

Since the angle of incidence is 90 degrees, the sine of the angle of incidence is 1.
1 / sin(r) = 1 / u

Simplifying the equation, we find:
sin(r) = u

Step 4: Calculating the Angle of Refraction:
To find the angle of refraction, we can take the inverse sine (or arcsine) of the refractive index.
r = arcsin(u)

Step 5: Determining the Light Ray Vector:
The light ray vector can be expressed in terms of the unit vectors i, j, and k. In this case, since the ray is incident along the y-axis, the initial light ray vector is given by:
R = j

To determine the light ray vector at the point where x = 1, we need to rotate the initial vector R by the angle of refraction r.
R' = cos(r) j - sin(r) i

Conclusion:
In conclusion, at the point where the x-coordinate becomes equal to 1, the light ray vector can be expressed as R' = cos(r) j - sin(r) i, where r is the angle of refraction and can be calculated using the relation u = 1 x^2.