Checking for parallel vectors - something is wrong.

[micheus]

Yeah. I know now.

For some reason I read and understood this text in the wrong way :

The dot product gives us a very nice method for determining if two vectors are perpendicular and it will give another method for determining when two vectors are parallel. Note as well that often we will use the term orthogonal in place of perpendicular.

Now, if two vectors are orthogonal then we know that the angle between them is 90 degrees. From (2) this tells us that if two vectors are orthogonal then,
...

ref. link

The Cross product have the behaviour I wanted:

Now, let’s address the one time where the cross product will not be orthogonal to the original vectors. If the two vectors, and , are parallel then the angle between them is either 0 or 180 degrees. From (1) this implies that,

ref. link

Sometimes, to learn, we need to know where we are going wrong. Not only that we are wrong.