Vector

A vector is an object with magnitude and direction, or an element of a vector space. It can represent quantities such as displacement and velocity, or serve as an abstract element in linear algebra.

Vector space: A set whose elements can be added and scaled while obeying the axioms of linear algebra. Vectors are the elements of this structure, which generalizes arrows beyond physical space.
Vector addition: An operation that combines vectors by adding their corresponding components. Adding displacement vectors yields the net displacement along a path.
Displacement: A change in position represented by a directed segment from an initial point to a final point. It is a direct geometric example of a vector, independent of the route taken.
Vector field: A function assigning a vector to each point in a space or region. A vector is one value; a vector field distributes such values across space.
Scalar: A single number that can scale a vector or combine with other numbers in algebra. Scalar multiplication changes a vector's magnitude without changing its direction, except for negative scaling.
Scalar multiplication: Multiplication of a vector by a scalar, scaling its components by the same factor. It changes length and can reverse direction when the scalar is negative.
Velocity: The rate of change of position with respect to time, including direction. Its direction distinguishes motion that speed alone cannot describe.
Tensor: A multilinear mathematical object that generalizes scalars and vectors across coordinate systems. Tensors extend vector-like quantities to objects with multiple input directions and output components.
Coordinate system: A convention for assigning numerical coordinates to points or directions in a space. Coordinates encode a vector as an ordered list relative to chosen axes.
Dot product: A scalar-valued operation on two vectors that combines their components and encodes their angular relationship. It measures alignment and supports projections and angle calculations.

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