What is the relation between vectors in physics and algebra? Vector math is something I find very interesting. However, we have never been told the link between vectors in physics (usually represented as arrows, e.g. a force vector) and in algebra (e.g. represented like a column matrix). It was really never explained well in classes.
Here are the things I can't wrap my head around:


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*How can a vector (starting from the algebraic definition) be represented as an arrow? Is it correct to assume that a vector (in a 2-dimensional space) $v = [1,1]$ could be represented as an arrow from the origin $[0,0]$ to the point $[1,1]$?

*If the above assumption is correct, what does it mean in the physics representation to normalize a vector?

*If I have a vector $[1,1]$, would the vector $[-1,1]$ be orthogonal to that first vector? (Because if you draw the arrows they are perpendicular).

*How can one translate an object along a vector? Is that simply scalar addition?


These questions probably sound really odd, but they come from a lack of decent explanation in both physics and algebra.

You can represent a vector as an arrow in cartesian coordinates by drawing an arrow from (0,0) to the vector (row vector) for example <3,2> taken as a point on the plane (3,4). In other words, your first bullet point is correct.
Vector normalization is the process by which one takes an arbitrary vector (a, b) and converts it to a new vector (a', b') where the length of (a', b') is 1.
For example, normalize(3, 4) = (3/5, 4/5). And we can verify that |(3/5, 4/5)| = sqrt(9/25 + 16/25) = sqrt(25/25) = 1.
That is correct. But keep in mind that for any vector (x, y), the vector (-x, y) will only be perpendicular to (x, y) if x = y.
