In inner product spaces, you can create an induced norm $||v||$ from the inner product by defining
$$||v|| = \langle v, v \rangle^\frac{1}{2}$$
But often (in proofs and whatnot) it's nicer to consider $||v||^2$. Why don't we define the induced norm as $||v|| = \langle v, v \rangle$?
Some reasons I could think of are:
1) Historically, this gives the straight-line metric for $\mathbb{R}^n$.
2) We are in some sense "double-counting" $v$ when we take $\langle v, v \rangle$, so we need to take the square root to "undo" it.
The triangle inequality fails if we tried defining the norm as $\langle v,v\rangle$. For instance, we would have, where $d(u,v)=||u-v||$, the distance function, on the vector space $\mathbb{R}^1$ for simplicity: $$d(0,1)=1$$ $$d(1,2)=1$$ $$d(0,2)=4>d(0,1)+d(1,2).$$