What Is the distance of three-dimensional coordinate $(x,y,z)$ from the origin $(0,0,0)$? Can I use the pythagorean theorem to fine the distance in three-dimensional coordinate also? That is, can i find it by $d = \sqrt{x^2+y^2+z^2}$ where $d$ is the distance between $(x,y,z)$ and $(0,0,0)$? 
I was having this question from studying the norm in Euclidean space. But I got curious since the norm of two dimensional vector $ \overline{x} = (x_1,x_2)$ is given by $||\overline{x}|| = \sqrt{x_1^2+x_2^2}$. I am curious if the concept of finding distance between the position of vector from the origin is what norm is. 
If it is not, how should I think of norm of a vectors in a geometry?
Also, for Euclidean scalar product $\langle \overline{x},\overline{y} \rangle = \sum\limits_{i=0}^{n}x_iy_i$, it seems like I am summing the areas of the square with sides $x_i$ and $y_i$ for each $i$. Is the right intuition for understanding the scalar product?
 A: Euclidean norm is one of possible norms. There are also the other ones, among others taxi and maximum. The Euclidean norms origins from the inner product, so $\Bbb R^n$ equipped with this norm is an inner product space. This is not the case for the above mentioned taxi and maximum norms. Of course, you quite the correct formula for this norm.
Thinking about a formula for a scalar product... for me this is not a proper way to understand it. The coordinates could be negative also. An inner product is a positive-defined symmetric bilinear functional (on a real inner product space, on a complex one this is more complicated). This formula directly comes from this definition if you know how to multiply the vactors of a canonical basis. So, another property of inner product is now useful: this is a product of norms of vectors and the cosine of the angle between them.
If some inner product is given, the Cauchy-Schwarz inequality allows us to introduce  the angle between two vectors.
A: Yes, the formula $d = \sqrt{x^2 + y^2 + z^2}$ would give you the distance from a point (x,y,z) to the origin. 
Norms are just functions which assign a "length" to a vector. You can think of this "length" as being a distance. There are many ways to measure "length", and there are more exotic norms which don't corresponds with the traditional notion of "distance".
Take a look at some examples of norms below.
https://en.wikipedia.org/wiki/Norm_(mathematics)#Euclidean_norm
