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:
- 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.