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High School Mathematics: You know... just because I've gotten weaker, it doesn't necessarily mean you've become any stronger, does it?

Haebom
I recently saw some news that left a strong impression on me. It was about matrices and vectors being excluded from some school curriculums. Looking closely, it’s not that these topics are being completely removed, but rather that they've become optional, making it possible to avoid geometry and vectors entirely if you choose. I ended up properly studying vectors at university while taking courses like discrete mathematics and engineering mathematics—almost being 'forced' to learn them. I think there’s a meaningful difference between learning this material with at least some background and trying to study it without any fundamental knowledge at all.
Vectors and vector databases have gained increasing importance, especially over the past few years alongside the emergence of Large Language Models (LLMs). These technologies provide new ways to process and analyze data, and their value is especially recognized in the fields of artificial intelligence and machine learning. In this article, we’ll explain what vectors and vector databases are, examine how they’re used, and take a closer look at why they matter.

What is a vector?

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A vector is a geometric object with both length (magnitude) and direction. For instance, it’s used to describe movement from one point to another in a two-dimensional space. Vectors can exist in various dimensions, with each dimension representing one element of the direction the vector points toward. These characteristics are built on a mathematical foundation known as vector algebra, which allows us to accurately analyze relationships between vectors.

How vectors are represented in code

In mathematics, a vector represents a quantity that has both magnitude and direction, and is used to describe things like movement in geometric space or forces in physics. Mathematical vectors can be defined in various dimensions, with each dimension representing a component of the direction in which the vector points.
In artificial intelligence—especially in machine learning and deep learning—vectors are used as a way to numerically represent data. For example, when handling text data, each word or sentence can be turned into a vector, making it readable and processable by a computer. These transformed vectors encode the semantic and contextual information of the words in numerical form.
There are various ways to represent vectors in code. The simplest approach is as an object storing the coordinates of its start and end points. However, since the key properties of a vector are its length and direction, we can also standardize all vectors to start from the origin (0, 0). In this case, storing only the coordinates of the end point is enough, simplifying the data structure.

Practical examples of using vectors (where are they used?)

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Computer graphics: In computer graphics, vectors are used to represent points in 2D or 3D space. These vectors are essential for determining pixel positions on the screen or for defining the location and direction of objects in 3D modeling.
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Recommendation engines: In recommendation systems, the attributes of items (for instance, a movie’s genre, length, and so on) are represented as multidimensional vectors. These are compared with the user’s preferences to make personalized suggestions. This process involves calculating the similarities between vectors to identify the most relevant items.
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Machine learning: In the field of machine learning, data in various forms—such as images, text, or audio—is converted into vectors to be processed. These vectors are then analyzed by algorithms for tasks like classification, prediction, or recommendation.
In fact, vectors have limitless potential applications, but recently, with the rise of techniques using VectorDB for optimizing and personalizing AI, the concept of vectors has become even more important.

The role of vector databases

A vector database is optimized for storing and retrieving vectors along with their associated metadata. These databases are designed to quickly find similar vectors by using algorithms such as Approximate Nearest Neighbor (ANN). This makes it possible to efficiently search for similar items within massive datasets.

How vector databases influence LLMs

Models like LLMs are limited in terms of input and output, but by using a vector database, you can overcome these limits by converting text and other kinds of data into vectors and storing them. This way, the model can retrieve and process only the most relevant parts of the data, greatly improving both efficiency and performance.
To sum up, vectors and vector databases are essential tools in modern software development—particularly in AI and machine learning. They offer powerful ways to model and handle data effectively, and the impact of this can be seen across a range of applications and systems. For anyone looking to use or study these technologies, understanding vectors and vector databases is incredibly important.
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