Vector (mathematics and physics)

In mathematics and physics, a vector is an object that has both magnitude and direction. It is typically represented geometrically as an arrow whose length is proportional to the vector's magnitude and whose direction points in the specified direction.

Mathematical Representation:

Mathematically, a vector is often represented as an ordered list of numbers, called components. For example, in a two-dimensional space (2D), a vector can be represented as (x, y), where x and y are the components representing the vector's displacement along the x-axis and y-axis, respectively. Similarly, in a three-dimensional space (3D), a vector can be represented as (x, y, z). The number of components corresponds to the dimension of the space in which the vector exists.

Vector Operations:

Vectors can be subjected to various mathematical operations, including:

• Addition: Adding two vectors results in a new vector whose components are the sum of the corresponding components of the original vectors.

• Subtraction: Subtracting one vector from another results in a new vector whose components are the difference between the corresponding components of the original vectors.

• Scalar Multiplication: Multiplying a vector by a scalar (a single number) results in a new vector whose magnitude is scaled by the scalar value, and whose direction remains the same if the scalar is positive, and reverses if the scalar is negative.

• Dot Product (Scalar Product): The dot product of two vectors results in a scalar value. It is calculated as the sum of the products of the corresponding components of the two vectors. Geometrically, the dot product is related to the angle between the vectors.

• Cross Product (Vector Product): The cross product of two vectors (in 3D space) results in a new vector that is perpendicular to both of the original vectors. The magnitude of the cross product is related to the area of the parallelogram formed by the two vectors.

Applications:

Vectors have wide-ranging applications in both mathematics and physics, including:

• Physics: Representing physical quantities such as velocity, acceleration, force, momentum, and electric and magnetic fields.

• Mathematics: Defining and manipulating geometric objects, solving systems of linear equations, and describing transformations.

• Computer Graphics: Representing positions, directions, and transformations of objects in 3D space.

• Engineering: Analyzing forces, stresses, and strains in structures.

• Navigation: Representing positions and headings in geographical coordinate systems.

Vector Spaces:

The set of all vectors of a certain type, together with the operations of vector addition and scalar multiplication, forms a vector space. Vector spaces are fundamental structures in linear algebra and provide a general framework for studying vectors and their properties.

Basis Vectors:

A basis for a vector space is a set of linearly independent vectors that can be used to express any other vector in the space as a linear combination. Common examples include the standard basis vectors in 2D (i = (1, 0), j = (0, 1)) and 3D (i = (1, 0, 0), j = (0, 1, 0), k = (0, 0, 1)).