Euclidean vector
Definition
A Euclidean vector (or simply a vector) is a geometric object that has both a magnitude (or length) and direction. Vectors can be added to other vectors according to vector algebra. A Euclidean vector is frequently represented by a line segment with a definite direction, or graphically as an arrow, connecting an initial point A with a terminal point B, and denoted by
.
History
The concept of the vector arose in the mid-19th century with the work of Irish physicist Hamilton and British mathematician Maxwell. Hamilton introduced the term "vector" in his work on quaternions, a number system that extends the complex numbers.
Properties
Vectors are defined in the Euclidean space, which is an extension of the ordinary Euclidean plane to any finite number of dimensions. A vector in Euclidean space is often represented by a tuple of real numbers, these numbers being the coordinates of the vector.
Magnitude
The magnitude of a vector is the length of the vector, which is a non-negative real number. It is denoted by ||v|| or |v|, where v is the vector.
Direction
The direction of a vector is the direction of the line segment that the vector represents. The direction can be defined by the angles that the vector makes with the axes of a coordinate system.
Zero Vector
The zero vector, denoted by 0, is a vector of zero length and arbitrary direction.
Operations
There are several operations that can be performed on vectors, including vector addition, scalar multiplication, dot product, and cross product.
Vector Addition
Vector addition is the operation of adding two or more vectors together. The result of vector addition is another vector.
Scalar Multiplication
Scalar multiplication is the operation of multiplying a vector by a scalar (a real number). The result of scalar multiplication is another vector.
Dot Product
The dot product, also known as the scalar product, is an operation that takes two vectors and returns a scalar. The dot product of two vectors is a measure of the extent to which the vectors are in the same direction.
Cross Product
The cross product, also known as the vector product, is an operation that takes two vectors in three-dimensional space and returns a vector that is perpendicular to both of the original vectors.
Applications
Euclidean vectors have many applications in various fields, including physics, engineering, computer graphics, and navigation.
In physics, vectors are used to represent physical quantities that have both magnitude and direction, such as force, velocity, and acceleration.
In engineering, vectors are used in the analysis of structures and mechanical systems.
In computer graphics, vectors are used to represent the position, direction, and color of objects.
In navigation, vectors are used to represent the position, velocity, and direction of moving objects.