TITLE: Squaring a Vector?
QUESTION [16 upvotes]: So this one is basic. And should be pretty quick. Lets say that I have a vector $\vec{r}$:
$\vec{r} = \vec{x} + \vec{y} + \vec{z}$
Is this true:
$\vec{r}^{2} = \vec{x}^{2} + \vec{y}^{2} + \vec{z}^{2}$
I know that you can't really multiply a vector by a vector in the normal sense. However you can take the dot product. In which case I think that this would hold. If I saw $\vec{r}^{2}$ would it be safe to read this as $\vec{r} \cdot \vec{r}$?
Then would $\vec{x}^{2} = \vec{x} \cdot \vec{x} = \left|\vec{x} \right|^{2}$
Making $\vec{r}^{2} = \left|\vec{x} \right|^{2} + \left|\vec{y} \right|^{2} + \left|\vec{z} \right|^{2}$
I have encountered strange notation where the square of a vector is written and I was wondering how to interpert it.

REPLY [4 votes]: May I suggest a good read on geometric algebra(outer products, r-blades ,...). Check out " An introduction to Geometric Algebra and Calculus " but essentially the answer to your question can be shown using the following definition
$$\vec{a}^2=\vec{a}\vec{a}=\vec{a}\cdot\vec{a}+\vec{a}\bigwedge\vec{a} $$
However 
$$\vec{a}\bigwedge\vec{a}=0$$
Hence
$$\vec{a}^2=\vec{a}\cdot\vec{a}=||\vec{a}||^2$$