I am trying to understand the vector norm. I have a few subquestions to the primary question here, what is the vector norm?
1. Firstly, lets take the Euclidean norm. Is then $\|x\|=d(x,0)$, where this is the Euclidean metric. Or do we allow vectors to sit anywhere in space, and $\|x\|$ is the length from the start of the vector to the end of the vector, sitting anywhere in space?
2. With the Euclidean norm, $\|x+y\|$ is the length of the vector $(x+y)$ right?
3. Let $X$ be a Euclidean space, If I have $\|x\|=1$, do I then have most of the vector, $x\in X$, , living inside of $S=\{a\in X| \quad\|a\|=1\}$, or do we treat the tip of the vector as the vector.
The confusion for $3.$ is that I was told to treat points in Euclidean space like vectors coming from the origin, and this has confused me since I first heard it.
I guess that's it, I can move forward for there I suppose and make a new question with follow ups.
Best Answer
A vector can be translated to anywhere since it is determined by its length and direction. So for example, the vector $(1,1)$ can start from the origin and end at the point $(1,1)$, or it can start from $(2,3)$ and end at $(3,4)$, whereas its vector form is always $(1,1)$.
However, when you say $d(x,0)$, it is a bit abuse of notation. Since this $x$ is actually the end point $x$ when the vector is starting from the origin.
Yes.
This is the vector $\vec{x}$, so it could be any vector that has length $1$. But you can also treat them all like they start from the origin.