I am trying to understand what it is the up vector in the ray tracing, but I'm not sure if I get it correctly. Is the up vector used to get the image plane? I will get the "right vector" making the cross product between the up vector and the forward vector (= target - origin). Does it have any other motivations? A good example could help me to understand it.
Let's revise how you construct these vectors:
First of all your forward-vector is very straightforward: vec3 forward = vec3(camPos - target) i.e. the opposite direction your camera is facing, where the target is a point in 3d space and camPos the current position of your camera.
Now you can't define a cartesian coordinate system with only one vector so we need two more to describe a coordinate system for the camera / the view of the ray-tracer. So let's find a vector that is perpendicular to the forward vector. Such vector can be found with: vec3 v = cross(anyVec, forward). In fact, you could use a random vector (except forward- and null-vector) to get the desired second direction but this is not convenient. We want that when looking along the z-axis (0, 0, 1) "right" is described as (1, 0, 0). This is true for vec3 right = cross(yAxis, forward) and vec3 yAxis = (0, 1, 0). If you would change your forward-vector so you aren't looking along the z-axis your right vector would change too, similar to how your "right" changes when you are changing your orientation.
So now only one vector is left to describe the orientation of our camera. This vector has to be perpendicular to the right- and forward-vector. Such vector can be found with: vec3 up = cross(forward, right). This vector describes what "up" is for the camera.
Please note that the forward-, right- and up-vectors need to be normalized.
If you would make a handstand your up-vector would be (0, -1, 0) and therefore would describe that you are seeing everything upsidedown while the other two vectors would be completely the same. If you look at the floor in a 90° angle your up-vector would be (0, 0, 1), your forward-vector (0, 1, 0) and your right-vector would stay at (1, 0, 0). So the function or "motivation" for the up-vector is that we need it to describe the orientation of your camera. You need it to "nod" the camera, i.e. adjust its pitch.
Is the up vector used to get the image plane?
The up-vector is used to describe the orientation of your camera or view into your scene. I assume you are using a view-matrix produced with the "look at method". When applying the view-matrix you are not rotating your camera. You are rotating all the objects/vertices. The camera is always facing the same direction (normally along the negative z-axis). For more information you can visit: https://www.scratchapixel.com/lessons/3d-basic-rendering/ray-tracing-generating-camera-rays/generating-camera-rays