Vector (mathematics and physics)

In mathematics and physics, a vector is a special kind of quantity that goes beyond just a single number. While a regular number, called a scalar quantity, can tell us how much of something there is, a vector can tell us both how much and in which direction. This makes vectors very useful for describing things like forces, velocity, and displacements, which all have a direction as well as a size.

Vectors were first used in geometry and mechanics to handle quantities that need both magnitude and direction. For example, when we push a box, the force we apply has a strength and a direction. Vectors help us work with these kinds of measurements in a clear and organized way.

In mathematics, vectors can also be thought of as tuples, which are fixed lists of numbers. Whether they come from geometry or from lists of numbers, vectors can be added together and multiplied by scalars (regular numbers). These operations led to the idea of a vector space, a set of vectors that follows certain rules. There are many different kinds of vector spaces studied in math, but the term "vector" usually refers to geometric vectors or lists of numbers.

Vectors in Euclidean geometry

Main article: Euclidean vector

In geometry, vectors help us describe directions and distances. Think of them like arrows that show both where something is and how far it is from a starting point. These arrows can be moved around without changing what they represent, which makes them very useful in many areas of math and science.

Vector quantities

Main article: Vector quantity

In mathematics and physics, a vector is like a special number that can show more than just a single value. It helps us understand things in the real world that need more information than just size, like direction as well. Vectors are important because they can be added together and changed by multiplying them with other numbers, which helps solve many problems in science and engineering.

Vector spaces

Main article: Vector space

In mathematics and physics, a vector is like a special kind of number that can show more than just size — it can also show direction. Vectors are used to describe things like force or speed in a particular direction. A vector space is a collection of these vectors where you can add them together or change their size by multiplying them by numbers, called scalars, while still staying within the same collection.

Vectors in algebra

Every algebra over a field is a special kind of space, but its pieces are not usually called vectors. Sometimes, they are called vectors because of old ideas from the past.

Some special kinds of vectors include:

• Vector quaternion, a type of number with no real part
• Multivector or p-vector, part of a bigger math idea called exterior algebra
• Spinors, also named spin vectors, used to understand turns and spins in space, linked to rotation vector and Clifford algebra
• Witt vector, a long list of numbers used for working with special number systems called p-adic numbers

Data represented by vectors

See also: Vector (data type)

Vectors can represent many different kinds of data. For example, a rotation vector shows the direction and angle of a spin. A Burgers vector describes changes in the structure of crystals. In music, an interval vector captures the distances between notes.

In statistics, a probability vector shows chances that add up to one, and a random vector groups together many changing values. A logical vector is simply a list of yes-or-no values.

Vectors in calculus

Calculus helps us study and work with vectors, especially in physics and engineering. It lets us understand motion in three-dimensional space by using vector-valued functions.

Vector calculus adds new operations like gradient, divergence, and curl to help solve real-world problems. We can use tools like line integrals to measure work along a path, surface integrals to find flow through a surface, and volume integrals to calculate things like mass distribution and charge density in three dimensions.