Vector Addition and Subtraction - interpretation If we have two vectors $a$ and $b$, both in $\Bbb R^n$, is it correct to think of 


*

*$a-b$ as how similar the two vectors are?

*$a + b$ as moving the vector $a$ in the direction of vector $b$?

 A: We can think of vector addition, $\vec a + \vec b$, as describing the point at the end of $\vec b$ if $\vec b$ started at the tip of $\vec a$.  In the picture below, the red vector is $\color{red}{\vec{a}}$, the blue vector is $\color{blue}{\vec b}$, and the yellow vector is $\vec a + \vec b$:

For vector subtraction, it is best to think of it with regards to addition.  First, realize that $-\vec a$ is just $\vec a$ reflected about the origin.  Then, treat $\vec{a} - \vec{b}$ as $\vec{a} + (-\vec{b})$ and visualize it as with addition.
If you're looking for a measure of vector similarity, we typically use the inner product or dot product: $\vec{a}\cdot \vec{b}$.  This makes since, because we'd like to think of "similarity" as a scalar quantity, rather than a whole vector.  If you've learned the relationship between the dot product and cosine, this makes a bit more sense:
$$\vec a \cdot \vec b = \|\vec a\|\|\vec b\|\cos\theta$$
...where $\theta$ is the angle between the two vectors, as in the image below:

So, vectors that point in the same direction are "similar," while vectors that point in opposite directions are "dissimilar."  Normalization is helpful here, because then $+1$ means "identical vectors," and $-1$ means "reflected vectors."  A value of $0$ means perpendicular. 
A: When I want to draw a representation of vector addition, $v+w$, and subtraction, $v-w$, I just remember that they are the diagonals of the parallelogram with sides $v$ and $w$:

Here's a way to visualize vector addition physically.  Consider a bird (or airplane) flying on a windy day.  If vector $v$ represents the velocity of the bird without the wind and $w$ represents the velocity of the wind, then $v+w$ is the actual velocity of the bird.  So just picture this: on a day with strong wind, the direction a bird is pointing will not necessarily be the direction its actually flying.
For vector subtraction, just think about displacements.  If you're at the head-end of vector $w$ and you move to the head-end of vector $v$, then the vector you move along is $v-w$.

I'm not sure how to quantify the similarity of two vectors, but for instance, you can quantify how parallel/perpendicular they are.
The closer to parallel that two vectors are, the closer to $\|v\|\|w\|$ that the dot product of $v$ and $w$ is.
The closer to perpendicular that two vectors are, the closer to $\|v\|\|w\|$ that the magnitude of the cross product of $v$ and $w$ is.
Thinking of vector addition as moving one vector in the direction of another is an interesting way to think about it.  And it works in $\Bbb R^n$, so if you want to think of it that way, there's no harm.  In fact, this idea of moving one vector along another is anticipating something you'll learn later called "parallel transport" -- though this concept really doesn't have much to do with vector addition in general.
A: *

*Yes, but this has to do with $R^n$ being a normed space rather than its algebraic properties. What this means is that in an abstract vector space you can't say what does it mean for a vector to be "small" unless you have a norm, or something similar.

*Yes.

A: Yes. Euclidean space generalizes naturally to n dimensions. 
Firstly, Euclidean space admits the L-2 metric, so when computing differences and additions, how "similar" they are or "how much you moved" depends on the distances of them you compute using the metric. 
Euclidean metric gives a natural measure of "distance" as we're used to in 1, 2, 3 dimensional space. On the other hand if you switch to a different metric space the meaning is lost. So, metric matters.
Secondly, you can see lower dimensional objects as projections of them sitting in higher dimension. 
Imagine a 3D cube, originally sitting in 5 dimensional space with axes v,w,x,y,z; but its length, width and height happen to coincide with axes x,y,z, so we can view the cube naturally in our natural frame of reference.
