When Numbers Become Directions
When Numbers Become Directions
Imagine someone asks where you are. You reply, "I'm 12. " Twelve what? Kilometres? Minutes? Floors? The answer is incomplete. A single number tells us how much. It doesn't tell us where. Now suppose you say, "I'm 12 kilometres north. " Suddenly, the answer is complete. You have described both magnitude and direction. Mathematics noticed that not every quantity behaves the same way. Some quantities answer only one question: How much? Temperature. Mass. Time. Money. These have magnitude but no direction. Mathematics calls them Scalars. Other quantities answer two questions. How much? In which direction? Walking 5 kilometres north is different from walking 5 kilometres south. A force of 100 Newtons pushing left is different from 100 Newtons pushing right. A car moving at 80 km/h east is different from one moving west. These quantities have both magnitude and direction. Mathematics calls them Vectors. Until now, vectors seem useful only for physics. But Machine Learning changes their meaning. A student's marks... A patient's medical report... A movie's attributes... A customer's buying behaviour... None of these have a physical direction. Yet they still become vectors. Why? Because in Machine Learning, direction doesn't mean north or south. It means how an object varies across its features. A student strong in Mathematics and Physics points in one direction. Another strong in Literature and History points in another. Different characteristics create different directions in a mathematical space. This is one of the most important ideas in AI. A vector is no longer just an arrow. It becomes the mathematical identity of an object. Every image... Every sentence... Every customer... Every song... Every face... Every protein... can be represented as a vector. Modern AI doesn't reason about objects. It reasons about vectors. Once every object becomes a vector, a new question naturally appears. How do we measure the similarity between two vectors? How do we know which vectors are close and which are far apart? The answer begins with one of the most fundamental ideas in linear algebra. Distance and Norms.