Considering Vectors Geometrically

Question

I have a few questions which a little research (searching the internet through Google) has not satiated. It seems that vectors are very important, even when considering them as the arrows which general physics textbooks do. For instance, they are very helpful when trying to prove geometric properties of, say, triangles.

At any rate, I have not been able to find any satisfying reading material that treats vectors as geometric arrows. I hope I am not being too vague; but as an example of one of my questions, does the geometric picture of vector addition follow from the way in which vectors are defined, or is picture built into the definition of a vector? I ask, because some sources I have read seem to suggest that it is built into the definition, while others do not.

The reason why I think I am having a fundamental difficulty with this, is that I am not a picture sort of fellow: I can not (very easily) deduce mathematical information from a picture, but prefer to start first principles when defining a concept, and then having the picture follow. Does this make sense? Is this a common problem?

Answer 1

Vector addition is defined to be component-wise. That meaning:

$$\textbf{x} + \textbf{y}=\begin{pmatrix}x_1&x_2&\cdots&x_n\end{pmatrix}+\begin{pmatrix}y_1&y_2&\cdots&y_n\end{pmatrix}=\begin{pmatrix}x_1+y_1&x_2+y_2&\cdots&x_n+y_n\end{pmatrix}$$

Now if you were to consider the geometric interpretation of these vectors in $\mathbb{R}^n$, since each of the components are added in the addition, you can liken it to placing the vectors end-to-end (i.e. adding each of the components of $\textbf{x}$ to $\textbf{y}$.

Answer 2

Since pretty much any finite vector space is isomorphic to $\mathbb{R}^{n}$ I like to think of vectors geometrically in this space for ease.

Now since our world is one that is spacial and temporal it is important to have I guess you would say concepts that we can use to talk about space. Just as a number is an abstract concept that can be used to talk about concepts that take up space, it is useful to have concepts that also can help us talk about direction of space, this is our vector.

In order to simplify the geometric interpretation of vector lets look at $\mathbb{R}^{2}$. This can be thought of looking at your self at a birds eye view. Now imagine at looking at someone standing still in this view and think of all the places he can go. Now the straight line paths that he would have to take to get to those places can be thought of vectors. They all have their specific direction and magnitude to them.

Just a little bit about span to maybe give a little more depth to the geometer of vectors. Well now lets say we want to create theories and properties of all vectors in this space well its pretty hard to do so if we have to look at every type of possible direction and magnitude to do so. But instead of looking at all the vectors we something quite interesting. We see if this person wants to travel 3 feet north he just has to walk 3 feet north and similarly if want to travel 4 feet west. But if he wants to travel 5 feet northwest he can either travel looking at the northwest direction and walk 5 feet or just travel 3 feet north then 4 feet west. thus the an place northwest of him can be traveled by and linear combination of north and west. It turns out he can actually travel anywhere with just some linear combination of north and west. These vectors that he can obtain by linear combination of north and west vectors is then called our span of these vectors. Thus what this does is makes it a lot easier in establishing properties of finite vector space such as $\mathbb{R}^{2}$ by just looking at scope of a span of some finite vectors

Now of course there is even more detail to these concepts that make them more rigors and abstract to apply to many different concepts but this is the gist of it

**NOTE this is just my naive interpretation so take with grain of salt if anyone has anything to correct or add please do so

Answer 3

This is an interesting book on the general topic of defining and understanding what a vector actually is.