When I did linear algebra in high-school, it wasn't of much importance where the vectors originated from and for me this is a really hard concept to grasp. It's like no matter where the two vectors are pivoted in 3-d space, their dot product is invariant.
Like, we don't even define an origin when talking about vectors... it's like they're freely floating in space. Why can we do this as in why do we not need to regard origin when we speak of vectors?
Is the vector attached to some object? like does it not matter where the 'tail' is.
Edit: this question arose mainly when I was learning about plotting vector fields, in that, I had to associate each point with a vector so definetely here the vectors origin is relevant but not in the previous case,why?
an extra part to the question:
I had also come across this problem when studying physics,
The person refers that cross product gives an axial vector. So I wonder how many types of vectors are there?
Does this mean that regular 'vector' that we learned of has many 'cousin-forms'? How many takes types of vectors are there? how do we distinguish between these kinds of vectors?
Summary: Why do sometimes care for where vectors pivot is from and sometimes not? and exactly how many kinds of vectors are there?