Assuming the concept of the camera used in this ray tracer is understood. Resulting from the view plane defined by the camera, and the chosen resolution our buffer for the final image has, we can calculate a two-dimensional array of vectors. The following diagram below illustrates the calculations necessary assuming a 2D scene. Applying the calculations in a 3D scene should only require minimal changes to the basic vector math at play and is therefore not elaborated on.
With w_vp being the width of the view plane and w_buff being the buffer width, we can calculate the direction each ray will take. Taking R_0 as an example then, we can then build a line using the vector equation. This line combined with a start- and endpoint will represent our ray. Assuming R_0 to be defined like shown below, we can calculate our first ray as such:
L_0 = (P_vp + T_0) + t * R_0
Now that we have a line equation, t is our way to define a starting point of our ray at t = 0 and an endpoint. Defining an endpoint is not strictly necessary. It only serves as an implementation detail for checking if a ray has ever hit anything or if it "left the scene" without an intersection. If so a background color or a default color will be written into the image buffer.
All that needs to be done now, is to calculate all intersections of each ray with the scene objects (Note: if multiple intersections occur, the intersection with the smallest value for t is selected). This will be the easiest for spheres as they are the simplest three-dimensional object imaginable (mathematically speaking). All we need to represent a sphere is a radius r and a position p(x, y, z).