TITLE: Why isn't 'intuitive' vector multiplication useful?
QUESTION [6 upvotes]: In Axler's Linear Algebra Done Right, in the introduction to vectors it says:

We could define a multiplication on $\mathbf{F}^n$ in a similar fashion [to addition], starting with two elements in $\mathbf{F}^n$ and getting another element of $\mathbf{F}^n$ by multiplying corresponding coordinates. Experience shows us that this definition is not useful for our purposes. 

In what ways was this definition tested? And how did it fail? Is there some fundamental reason why this definition of multiplication is bad for a vector space?
Are there any examples of successful application this definition of multiplication?

REPLY [2 votes]: The dot product doesn't change when you rotate the vectors. Proof: It satisfies $\|a+b\|^2=\|a\|^2+2a\cdot b+\|b\|^2$, and the magnitude function is invariant under rotation.
On the other hand, this (let's call it $\otimes$) does. Proof: $(1,0)\otimes(0,1)=(0,0)$, but if you apply a forty-five degree rotation, you get $(\frac{\sqrt2}2,\frac{\sqrt2}2)\otimes (-\frac{\sqrt2}2,\frac{\sqrt2}2)=(-\frac12,\frac12)\ne(0,0)$.