In the book Calculus Vol 1- Tom M. Apostol .Before beginning to define the dot product of two vectors he tells
It can be shown that except $n=1, 2$, it is not possible to introduce multiplication in $V_n$ so as to satisfy all the field properties.
Which I think might be a motivation to defining dot product and cross product(for $n=3$).
I know that
For $n=1$ we can simply define multiplication as $\left(a_1\right)\times\left(a_2\right)=\left(a_1a_2\right)$
And for $n=2$ $\left(a_1,b_1\right)\times\left(a_2,b_2\right)=\left(a_1a_2-b_1b_2,a_1b_2+a_2b_1\right)$
I can't really follow why similar definition to $n>3$ creates problems
Can anyone explain the proof?. If there are better ways that served as motivation please explain/share link.
Thank you.
The reason why there aren't '3-dimensional numbers' (or higher) is deep and hard. Mathematicians looked for such things for a long time.
But so that you have a concrete reference, what you are interested in is the idea behind the proofs of the Frobenius Theorem, which says that the only multiplications that behave mostly like what we think of as multiplication are the reals and the complexes (and the quaternions, if we don't care for commutativity) or Hurwitz's Theorem, which essentially says the same thing but with a different background.
One way to think of it, maybe, is to generalize the idea that multiplication of complex units lead to rotations on the unit circle. Similarly, multiplications of real units lead to degenerate rotations on the degenerate 1-dimensional unit circle. So you might think that higher dimensional multiplication of units would also have to do with rotations on a higher dimensional sphere. But as soon as you go past 2-dimensions, rotations don't commute. It is possible for $r_1 \circ r_2$ to be different than $r_2 \circ r_1$, where $r_i$ are rotations of a sphere. What this means is that we might not expect commutativity, to say the least.
It should also be pointed out that the 4-dimensional number system, the quaternions, are naturally linked to spheres and rotations, and in fact do not have commutativity. I cannot think of another heuristic to explain why the octonions lose associativity, though.