Definition of a Vector

BASIC DEFINITION

A vector is a mathematical object determined by direction and magnitude (length). It is used to describe a directed quantity – meaning that in addition to a value (length), it also contains information about the direction in which it acts. In a plane or in space, a vector is often represented as a directed line segment with an arrowhead.

From a mathematical viewpoint, a vector is defined as an ordered pair (in a plane) or an ordered triple (in space) of real numbers, which represent the coordinates of its orientation relative to an origin.

• In a plane, we denote it as v = (x, y).
• In space, as v = (x, y, z).
  Here, x, y, and z are the components of the vector.

COMPONENTS OF A VECTOR

• INITIAL POINT (from which the vector originates, if representing displacement between two specific points).
• TERMINAL POINT (where the vector points, if representing displacement).
• COMPONENTS – If a vector represents the displacement from point A(x₁, y₁, z₁) to point B(x₂, y₂, z₂), its components are (x₂-x₁, y₂-y₁, z₂-z₁). Often, vectors are considered from the origin (0,0,0), in which case their components are the coordinates of their terminal point.
• MAGNITUDE (also norm or length), denoted by |v|.

The magnitude of a vector v = (x, y) in a plane is calculated as:
|v| = √(x² + y²)

And for a vector v = (x, y, z) in space:
|v| = √(x² + y² + z²)

CALCULATION EXAMPLE

Let the vector a = (3, 4) be given. Its magnitude is calculated as:
|a| = √(3² + 4²) = √(9 + 16) = √25 = 5.

This means that vector a has a magnitude of 5 units and, if starting from the origin, is directed towards the point (3, 4).

CONCLUSION

This mathematical object is fundamental for describing direction and magnitude in many geometric and algebraic constructions. It is composed of components along each coordinate axis, which allows it to be handled in a plane or space using standard arithmetic operations such as addition, scalar multiplication, and calculation of its length.