What is the relation between vectors in physics and algebra?

Question

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.

Answer 1

Physicists tend to emphasize a geometric interpretations of vectors, which mathematicians need not do. This is because one of the main uses of vectors in physics is to talk about the geometry of some system, while vectors in general can be used in more abstract senses (and perhaps with no geometric interpretation at all).

Concerning your first bullet point, your interpretation is correct: this is a model of a vector space as an algebra of directions. Each arrow has a starting point and an ending point, which determines both length and orientation. These are the properties of a "direction".

Normalizing vectors is a way to lose the length information while maintaining directionality; it makes vectors unit length.

Yes, you're correct in saying those vectors would be orthogonal. This is part of the geometric interpretation.

I'm not sure what you refer to by "translating an object about a vector". It could be taking the location of the object and simply moving every point in the direction of the vector. This is better described by vector addition.

Answer 2

Actually, I think your intuition is very good here. An "arrow" has two bits of information: its length and direction. When you have a vector $v = [a, b]$, it's describing an arrow that moves $a$ in the horizontal direction and $b$ in the vertical direction. Since this defines a right triangle, this can be converted into length and direction information.

Normalizing a vector simply means adjusting the length of a vector to be $1$ without changing its direction.

To address your orthogonality question, yes, as you mentioned, $[-1, 1]$ and $[1,1]$ are indeed orthogonal, because of the geometric reasoning you correctly described. However, they are only orthogonal with respect to the standard inner product. This is imply the dot product, which has a lot of geometric connections. However, in mathematics, we can define many other types of inner products that don't have as nice geometric connections.

Answer 3

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.