The meaning of the equation of the line that passes through a point and intersects with a vector

Question

I have a question about the meaning of a system of equations represented in R3 (3 dimensions if I'm not mistaken), in which we are finding the intersection of vector u (equation 1) with the line parallel to that vector and passing through point V. So far so good, I think anyone familiar with geometry and vectors will understand that. What I can't quite grasp and find extremely difficult is why in the system of equations we suddenly define equation 1 as it is, instead of using a general equation of a line (Ax + By + Cz + D = 0), and how we calculate a vector perpendicular to u when we are multiplying x and z (which is how we define the axes of vector u). In other words, from my perspective, we are multiplying a vector by itself, so how can we calculate a parallel vector if all we are doing is multiplying two vectors? My reasoning and way of looking at it are probably incorrect, so if you don't understand something, you can try to think about it your way, and if you come up with an understanding of the equation and why everything is the way it is, I kindly ask you to share it. I apologize for any inconvenience and for taking up your time, but hopefully, we can learn from this, especially me due to my lack of knowledge, and figure out how to solve it. Below, I've included images of the system and what we're doing, with the part I don't understand also highlighted in red.