All of this is true because Quaternions are the origin of modern vector analysis.
Hamilton, which is the inventor of quaternions, is also responsible for
1. The definition of "vector".
2. The definition of "Scalar Product"
3. The definition of "Cross Product"

Oliver Heaviside and Josiah Willard Gibbs, who are the forefathers of modern vector analysis, adopted the vector part idea, which came from Quaternions, and disregarded the scalar part.

more interesting info can be seen at:
History of Quaternions by Sweetster
History of Quaternions in Wikipedia