Defining a displacement vector not touching the origin

[Hijaz Aslam]

In the diagram let the magnitude of the vector BC and OA are 'x'. I am confused with this part. Vectorially we don't say that the vector BC is $-x$, because $-x$ is represented by OA. Then how do we represent BC?

It's said a the position of a vector doesn't matter, I mean one can shift it in any way unless their magnitude and direction are changed, is it applicable to a displacement vector? As in the case above we just can't shift the vector $-x$ to BC and call it $-x$. For instance in the case of a ball thrown from a tower. If we take the origin as the top of the tower, we can always say the last $x$ meters of the tower. Am I confusing any concept?

 

[HallsofIvy]

Yes, you are confusing "vector" with a specific interval, "the last x meters of a tower".