What is the geometric interpretation of a vector squared? I'm working through Introduction to Space Dynamics by William Tyrrell Thomson. I am having to do a lot of research to make it through even small parts, but I am unable to find information to make me confident enough to solve this question from the book:


  What is the geometric interpretation of $\left(\vec{a} + \vec{b}\right)^2$?


To start, I'm considering a simplified form: $\vec{c}^2, c = \vec{a} + \vec{b}$

This is where I get stuck, as I have not been able to find how to handle a vector multiplied by itself. Information one place states that a vector multiplied by itself is the same as the dot product of a vector with itself: $\vec{c}\cdot\vec{c}$. Other places I've found information which makes me think that multiplying a vector by another vector in the sense one would multiply a scalar by a scalar is not a valid operation to perform.

Which of these two is the case, or is it a third case I haven't considered?
Am I approaching the problem incorrectly?
$\vec{c}^2$ is an alternative notation for $\vec{c}\cdot\vec{c}=||\vec{c}||^2$

A motivation for this is that the formulas 

$$||\vec{a}+\vec{b}||^2=||\vec{a}||^2+2\vec{a}\cdot\vec{b}+||\vec{b}||^2$$

$$||\vec{a}-\vec{b}||^2=||\vec{a}||^2-2\vec{a}\cdot\vec{b}+||\vec{b}||^2$$

and

$$(\vec{a}+\vec{b})\cdot(\vec{a}-\vec{b})=||\vec{a}||^2-||\vec{b}||^2$$

now becomes

$$(\vec{a}+\vec{b})^2=\vec{a}^2+2\vec{a}\cdot\vec{b}+\vec{b}^2$$

$$(\vec{a}-\vec{b})^2=\vec{a}^2-2\vec{a}\cdot\vec{b}+\vec{b}^2$$

and

$$(\vec{a}+\vec{b})\cdot(\vec{a}-\vec{b})=\vec{a}^2-\vec{b}^2$$

Similarities with the corresponding scalar identities make them easier to remember.
I think you are wise to focus on 
$\vec{c} \cdot \vec{c}$, rather than $(\vec{a} + \vec{b})^2$.

Let $\vec{c}=(2,1)$. Then 
$$\vec{c} \cdot \vec{c} = (2,1) \cdot (2,1) = 2 \cdot 2 + 1 \cdot 1 = 5 \;,$$
which is the square of the length of the vector $(2,1)$,
i.e., that vector has length $\sqrt{5}$.
The dot-product multiplication is "component-wise," i.e., product of $x$-coords,
plus product of $y$-coords.
So the geometric interpretation is: the square of the length of the vector.

(There is another prominent vector multiplication, the cross product.)