The Powerful ``Directed Line Segment''

A vector, also known as a directed quantity, is defined as a quantity in a linear space that requires both magnitude and direction to be fully represented. For us, however, it is more appropriate to use the simplest concept: a vector is a “directed line segment.” The concept of vectors originates from physics, yet it is not applied solely within physics. The emergence of vectors provided a powerful tool for the development of geometry and physics. There is a saying: “The invention of logarithms lengthened the life of astronomers.” I can say without exaggeration that the development of vectors is also continuously lengthening the lives of mathematicians and physicists!

Why do vectors possess such immense power? In fact, when we study mathematics, we always consciously or unconsciously lean towards algebra, which involves only calculation, and often feel a bit of “repulsion” towards geometry, which requires “drawing.” Over time, we become more willing to engage with algebra rather than geometry. This is because we believe algebra consists of quantitative calculations, while geometry requires abstract thinking and imaginative abilities. It is precisely because of this that vectors have been endowed with vitality. Since we like calculation, we turn geometry into something calculable; we do not just want to calculate algebra, we want to “calculate geometry”!

How can geometry be transformed into a calculable problem? Analytic geometry was born for this purpose. It places all geometric problems into a coordinate system and equates certain mathematical equations (such as the equality of slopes of two lines) with certain geometric relationships (such as two lines being parallel). In this way, by proving the mathematical equation through calculation, we also prove the geometric relationship.

At that time, Descartes stood at the height of the natural philosophy of methodology, believing that the geometry of the Greeks relied too heavily on figures and constrained human imagination. Regarding the algebra popular at the time, he felt it was entirely subordinate to rules and formulas and could not become a science for improving the intellect. Therefore, he proposed that the advantages of geometry and algebra must be combined to establish a “true mathematics.” The core of Descartes’ thought was to reduce geometric problems to algebraic forms and use algebraic methods for calculation and proof, thereby achieving the ultimate goal of solving geometric problems. Based on this idea, he founded what we now call “Analytic Geometry.”

However, this coordinate system method has certain limitations (though its power is immense); it only expresses “quantity” and does not express direction. That is to say, one must use two pieces of data to determine a quantity. Most things in physics carry a “direction” (such as force, velocity, and acceleration). It is not just physics; even in mathematics, considering certain mathematical quantities along with their direction makes things much simpler. Vectors are based on certain definable operations (which have corresponding meanings in physics) and use analytical methods to yield surprising results!

I often say to the students around me: “Vectors are for people with poor spatial thinking skills. I am so clumsy, so I must master vectors thoroughly.” This sentence carries a bit of a joke, but what it says reflects a fact: using vectors can reduce abstract geometric thinking, thereby accelerating the pace of our research—which is why it lengthens life! Therefore, we can say, “Algebraists are not necessarily proficient in geometry; but geometers are certainly proficient in algebra—because they are proficient in vectors!”

The above are just a few of my reflections when I first started encountering vectors. As for examples of the role of vectors, they are not within the scope of this article. In future articles, I will write more examples of using vectors to solve problems in geometry, physics, and astronomy, allowing everyone to experience the power of vectors more deeply.

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