Vectors

Vectors are quantities characterized by both magnitude and direction. In a two-dimensional or three-dimensional space, a vector is represented as an arrow of a specific length, starting from an initial point (O) and ending at a point (x, y).

A free vector (or simply a vector) is a set of equivalent directed segments.

Two or more directed segments are considered equivalent when they share the same magnitude, direction, and orientation.

• Magnitude refers to the length of the vector. It is always a positive value expressed in a specific unit of measurement.
• Direction indicates the line along which the vector lies.
• Orientation defines the vector's direction, represented by an arrow.

The study of vectors in two or more dimensions is part of vector calculus, a branch of linear algebra that focuses on vector quantities.

What are vector quantities? Vector quantities are those that can be described using a magnitude (length or intensity), a direction, and an orientation. They differ from scalar quantities, which can be expressed as real numbers with a unit of measurement.

An Example of a Vector

The directed segments u, v, and w belong to the same free vector as they have the same direction, orientation, and magnitude (length).

They form part of the same equivalence class.

The only distinction between the three directed segments is their starting point.

Position Vector and Displacement Vector

Several concepts are associated with vectors, including the following:

• Position Vector. The position vector defines the location of a point in space relative to a reference system.
  
• Displacement Vector. A displacement vector is depicted as an arrow originating at the initial position of a point and pointing to its final position. It represents the change in the position vector.
  

Vector Operations

Vectors can undergo various operations, such as addition, subtraction, and scalar multiplication.

Each operation has specific rules and properties that define the magnitude and direction of the resulting vector.

The primary operations involving vectors include:

• Vector Addition
  If (x₁, y₁) and (x₂, y₂) are the endpoints of two vectors, u and v respectively, then (x₁ + x₂, y₁ + y₂) is the endpoint of the vector u + v.
  
  The sum of two vectors in the space R² = RxR can also be determined using the parallelogram method.
  
• Scalar Multiplication
  If (x, y) is the endpoint of the vector v, then the product kv results in a vector with the endpoint (kx, ky).
  
  The product of a real number k and a vector v is obtained by multiplying the magnitude of v by k. The direction remains the same if k > 0, or is reversed if k < 0.

Vectors in Higher Dimensions

A multidimensional vector is essentially an ordered list of n values, which can be conceptualized as an n-tuple of real numbers. Each value is denoted by a symbol with a subscript that indicates its position in the list, such as x₁, x₂, ..., x_n.

Vector notation typically uses a letter with an arrow on top, represented as:

$$ \vec{v} = (x_1, x_2, ..., x_n) $$

Vectors are essential for representing quantities that cannot be described by a single value. In physics, force is a practical example of a vector quantity.

Note. On Okpedia, vectors are distinguished from scalars by displaying them in bold. For example, u, w, and v are vectors, while u, v, and w represent scalars.