Scalars and Coordinate Systems
Consider a collection of particles arranged on a grid. Each particle is labeled by its mass. Using coordinate axes, we can specify the position of any particle by a pair of numbers such as (x, y). The mass at a given point can be written as M(x, y).
If we rotate or shift the coordinate system, the numerical coordinates of the particle will change. For example, the same particle may now be described by new coordinates (x′, y′). However, its mass remains exactly the same.
Mass does not change under a coordinate transformation.
Definition
A scalar is a physical quantity that remains unchanged under coordinate transformations.
Mass is not the only example. Temperature, energy, and speed are also scalar quantities because their values do not depend on how we choose our coordinate system.
Why Scalars Are Not Enough
Although scalars fully describe some physical properties, they are not sufficient to describe quantities that involve direction. For example, the direction of motion of a particle or the direction of a force acting on it cannot be represented by a single number.
To describe such directional quantities, we need vectors. Just as scalars are defined by their invariance under coordinate changes, vectors are defined by how they transform when the coordinate system is rotated or shifted.
Example 1 — Example: Mass vs. Force
Mass remains the same regardless of coordinate rotation, so it is a scalar. Force, however, changes its components when the coordinate axes rotate, so it must be described as a vector.