Scalars and Vectors: Definitions, Representations, and Properties

Definition of a Scalar: A scalar is defined as a physical quantity that possesses magnitude only and has no associated direction.
Components of Magnitude: Magnitude consists strictly of a numerical value accompanied by an appropriate physical unit (a number with a proper unit).
Algebraic Behavior: Scalar quantities are manipulated using ordinary arithmetic operations and obey simple algebra (scalar addition, subtraction, multiplication, and division).
Classification of Electric Current: Electric current is classified as a scalar quantity because it obeys the rules of simple algebra rather than the laws of vector addition, despite having an associated flow direction in electric circuits.
Examples of Scalar Quantities:
- Mass
- Time
- Temperature
- Distance
- Speed
- Energy

Symbolic Representation:
- Scalars are symbolically represented using ordinary italic letters, such as $a$ , $b$ , and similar standard alphabetic characters.
- They do not require any specialized or directional diacritical notations (such as arrows or boldface).
- Symbolic Examples:
- Mass: $m = 5\,\text{kg}$
- Temperature: $T = 300\,\text{K}$
Graphic Representation:
- To represent a scalar graphically, a proper scale is established by choosing a standard conversion factor so that the measured physical quantity and its units can be conveniently drawn on paper.
- Graphic representation of a scalar requires no specification of spatial orientation or directional coordinates, as the magnitude fully specifies the quantity.

Definition of a Vector: A vector is defined as a physical quantity that possesses both magnitude and direction.
Governing Rules: In addition to possessing magnitude and direction, a quantity must strictly obey the laws of vector algebra (such as vector addition, vector subtraction, and vector products) to be classified as a true vector.
Examples of Vector Quantities:
- Displacement
- Velocity
- Acceleration
- Force