Normalizing Vectors to get short numbers

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$\vec{A}$ is vector agent, $\vec{O}$ is vector Object, $m$ is a constant integer.

The following expression is repeated with a different O for every loop cycle:

$$\vec{A_{a}}=m\vec{A_{a-1}}+\vec{O_a}$$

The elements of $\vec{A}$ and $\vec{O}$ are real numbers and after $a$ loop cycles the numbers become very big in terms of magnitude and significant figures. This means i must normalize the result on each loop cycle. When using $$\vec{A_{a}}=\frac{m\vec{A_{a-1}}+\vec{O_a}}{|m\vec{A_{a-1}}+\vec{O_a}|}$$ to normalize the result, the numbers are still long.

The problem boils down to finding a number, that when it divides all elements of A yields the shortest/nicest numbers. Is there such thing? If not, what would be the best way to simplify this vector? I am not concerned with the vector's magnitude- only its direction, while maintaining computability of the numbers.