Confused by more definitions of a vector

Question

Recently I am going through some basic concept of vectors and I am watching a video in which a prophessor says that a vector is not a point in three dimentional space but actually it is the length of the line from the reading of the vector's coordinates to an origin. Or is it the origin? After all we both essentially meant an ordered $3$-tuple $(0,0,0)$ - Essentially the magnitude of an auxiliary displacment vector.

So, is this number the vector we began talking about?

My confusion is: If every vector is joined or connected with an origin then how did I correctly learn and understand the parallelogram law? (Perhaps I believe my own eyes. Perhaps it is valid to interpret the coordinates relative to $(0,0,0)$ in the process of relating to another vector).

Another thing is that he says a vector has no position but instead it has a direction and a magnitude. But in my point of view $(x,y,z)$ need not be the vector's position in the first place..

Answer 1

A vector has a direction and a length.

It is not located at a particular position in space and can translate freely.

You can specify it by providing the coordinates of two endpoints, and its components are the pairwise differences between theses coordinates.

$$(x,y,z):=(x_e-x_s,y_e-y_s,z_e-z_s)$$

If the starting endpoint is the origin, then the components are equal to the coordinates of the ending endpoint.

$$(x,y,z):=(x_e-0,y_e-0,z_e-0)$$

You can add a vector to a point, giving a point. You can add a vector to a vector, giving a vector.

As the coordinates of a point and the components of a vector are denoted similarly, some kind of (harmless) confusion is made between a vector and the point reached from the origin.

Answer 2

Pretty much correct. Where it is, where it directs and what is it's size depend (more often than not) on how you choose to transform and look at the world (questions).

Vectors transform. "Scalars" pretty much remain the same.

Furthermore, this is a kind of a pure geometric perspective to begin with. What about algebra?

Sometimes it is an aside algebraic or techincal need to have extra structure and has nothing to do with the above, as well.

And finally: It would be wrong to adopt different stuff named vector. It's the same object. Simply a great abstraction.

Answer 3

I would like to reflect only this part of the question:

"if every vector is joined or connected with origin then how we can add up any two vector with parallelogram law?"

The following figure explains that we don't need vectors to be placed at the ending points of vectors just because we want to use the parallelogram law.

As shown, we need only parallel lines...