Summary: Not only was the *first* product of vectors defined exactly this way, but the dot and cross products are just *parts* of this first product -- the product of quaternions AKA the Hamilton product.

You might be interested to know that $i$, $j$, and $k$ were introduced by William Hamilton as the imaginary parts of quaternions. And he developed the rules for multiplying his numbers with exactly that distributive property (and some other nice properties, like associativity) in mind.

For some background, recall that complex numbers are written in Cartesian form as $a+bi$, where $a, b$ are real numbers and $i$ is the imaginary unit such that $i^2=-1$.

Well, complex numbers are *extremely* useful in physics, but Hamilton wanted to go one step further -- he wanted to generalize complex numbers to $3$ dimensions. Then we could use these generalized complex numbers to represent points in $3$-D space.

He never found a $3$-D generalization of complex numbers but he did find a $4$-D generalization -- the quaternions.

Quaternions are numbers of the form $a+bi+cj+dk$ where $a, b, c, d$ are real numbers, and $i, j, k$ are *distinct* imaginary units. These imaginary units have definitions similar to the imaginary unit in the complex numbers: $i^2=j^2=k^2=-1$. But you also need some way of figuring out what things like $ij$ means. To this end, Hamilton figured out you only need to add one more rule $ijk=-1$. Then you can figure out what $ij$ is by multiplying both sides of $ijk=-1$ by $k$. Then you get $$(ijk)k = (-1)k \\ ij(k^2)=-k \\ ij(-1) = -k \\ ij = k$$

Hamilton is also the guy who introduced the words "vector" and "scalar". To Hamilton scalars were the real parts of his quaternions and vectors were the imaginary parts. For instance, if you had $z= 4 + 5i + 6j -7k$, then the scalar part of $z$ is $4$ and the vector part is $5i+6j-7k$.

Quaternions are not only the progenitors of modern vectors, but also of the dot and cross products. Let's look at the **Hamilton product** of two quaternions with 0 scalar parts:

$$(a_1i + a_2j + a_3k)(b_1i+b_2j + b_3k) \\ = a_1b_1(ii) + a_1b_2(ij) + a_1b_3(ik) + a_2b_1(ji) + a_2b_2(jj) + a_2b_3(jk) + a_3b_1(ki) + a_3b_2(kj) + a_3b_3(kk) \\ = -a_1b_1 + a_1b_2(k) + a_1b_3(-j) + a_2b_1(-k) - a_2b_2 + a_2b_3(i) + a_3b_1(j) + a_3b_2(-i) - a_3b_3 \\ = -(a_1b_1 + a_2b_2 + a_3b_3) + \left[(a_2b_3-a_3b_2)i + (a_3b_1-a_1b_3)j + (a_1b_2 - a_2b_1)k\right]$$

But you should recognize this part part: it's just $-\mathbf a \cdot \mathbf b + \mathbf a \times \mathbf b$. After some study it was discovered that these two *subproducts* of the Hamilton product are very, very useful.

You may be wondering why we don't use quaternions nearly as much as vectors today -- it's mostly due to the work of some very influential mathematicians and physicists like Oliver Heaviside and Josiah Gibbs.

In particular, Josiah Gibbs really did not like Hamilton's approach to quaternions. So he spent several years developing a calculus on *just* the vector parts of quaternions -- which he saw as much more useful than the full quaternion analysis.

Gibbs produced a short treatise on his methods but was unable (or unwilling) to write up a full textbook on the subject. A new graduate student from Harvard, Edwin Wilson, was convinced by his advisor that *he* should write a textbook based on Gibbs' treatise. It was entitled *Vector Analysis* and it was one of the most important textbooks in the history of mathematics. Not because it was so visionary per se, but because the methods, notations, and ideas of this book have become so fundamental to modern mathematics that we teach it to high schoolers (and of course, every university student was has to take vector calculus, linear algebra, or classical mechanics -- so pretty much *all* of them).