Vectors

A vector is a mathematical entity that has both a magnitude (or size) and a direction. Vectors are used in many areas of mathematics and physics to represent quantities such as velocity, force, and displacement.

Basic Concepts

Magnitude: The magnitude (or length) of a vector is a non-negative number that describes the extent of the vector in space. It is typically denoted by vertical bars (|v|) around the vector symbol.
Direction: The direction of a vector is the orientation of the vector in space. It is typically defined in terms of angles with respect to a reference axis.
Components: A vector can be represented by its components along the axes of a coordinate system. For example, in two dimensions, a vector v can be represented as v = (v_x, v_y), where v_x and v_y are the x and y components of the vector.

Vector Operations

Several operations can be performed with vectors:

Vector Addition: Vectors can be added together to produce a new vector. The components of the resulting vector are the sums of the components of the original vectors.
Scalar Multiplication: A vector can be multiplied by a scalar (a real number) to produce a new vector. The components of the resulting vector are the components of the original vector multiplied by the scalar.
Dot Product: The dot product (or scalar product) of two vectors is a scalar whose value depends on the magnitudes of the vectors and the angle between them.
Cross Product: The cross product (or vector product) of two vectors is a vector whose direction is perpendicular to the plane containing the original vectors, and whose magnitude depends on the magnitudes of the vectors and the sine of the angle between them.

Applications

Vectors have numerous applications in physics, engineering, computer science, and other fields. They are used to represent physical quantities such as force, velocity, and acceleration, which have both a magnitude and a direction. In computer science, they are used in graphics and machine learning, among other areas.