Vectors and Scalars: Magnitude and Direction Defined A vector is characterized by both magnitude and direction, while a scalar only provides magnitude. The distinction is clear when considering that ‘5 miles per hour’ lacks direction and is merely a scalar, whereas ‘5 miles per hour east’ is a vector known as velocity. Understanding this difference lays the groundwork for visualizing vectors beyond plain numerical speed.
Coordinate Representation and Multi-Dimensional Extension Vectors can be visualized as arrows where the length indicates magnitude and the arrow’s angle shows its direction. They are mathematically represented as coordinate pairs, such as (5, 0) for horizontal movement or (3, 4) for a vector whose magnitude, calculated using the Pythagorean theorem, equals 5. This coordinate notation not only simplifies computations in two dimensions but also extends to higher-dimensional spaces in linear algebra.