EUCLIDEAN VECTOR Meaning and
Definition
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An Euclidean vector, also known as an ordinary or geometric vector, is a fundamental mathematical object within linear algebra and geometry. It represents a quantity that encompasses magnitude and direction in a Euclidean space, a geometric construct with three dimensions.
Defined as an ordered pair of points in Euclidean space, an Euclidean vector consists of both a starting point, often referred to as the origin, and an endpoint. The difference between these two points determines the vector’s position and direction. Moreover, each component or coordinate of the vector, typically represented by letters, corresponds to the difference between the corresponding coordinates of the starting and ending points.
Euclidean vectors possess two principal properties: magnitude and direction. The magnitude, or length, of a vector is determined using the Pythagorean theorem, which indicates that the squared sum of the coordinates yields the sum of squares for the vector’s magnitude. Consequently, vectors are not arbitrary shapes or figures, but instead are mathematical entities defined by their length and orientation relative to the coordinate system.
Representing physical quantities such as velocity, force, or displacement, Euclidean vectors are fundamental tools in mathematics and physics. They find numerous applications in various fields, including engineering, computer graphics, and robotics. In addition, Euclidean vectors serve as a basis for more advanced vector spaces and mathematical concepts, extending beyond the classical three-dimensional Euclidean space to higher dimensions and abstract spaces.
Etymology of EUCLIDEAN VECTOR
The term "euclidean vector" derives from the word "Euclidean" and the concept of vectors in Euclidean geometry.
Euclidean refers to Euclid, a Greek mathematician who lived around 300 BCE and is often considered the father of geometry. Euclid's work, called "Elements", established the foundation of geometry as we know it and introduced many fundamental concepts, including vectors.
A vector in Euclidean geometry refers to a geometric object that has both direction and magnitude. It can be represented as an arrow or line segment with a specific length and direction in space. Euclidean vectors are extensively used in various fields like physics, engineering, and mathematics to describe physical quantities, forces, and displacements.
Therefore, the term "Euclidean vector" combines the name of Euclid with the concept of vectors in his geometry, highlighting their application and significance in Euclidean space.
