Vector (mathematics and physics)

In mathematics and physics, a vector is a special kind of quantity. It is more than 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. They help us describe things like forces, velocity, and displacements. These all have a direction as well as a size.

Vectors were first used in geometry and mechanics. They help handle quantities that need both size 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 measurements in a clear way.

In mathematics, vectors can be thought of as 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. This is 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 where something is and how far it is from a starting point. These arrows can be moved around, but they still mean the same thing. This makes them very useful in many areas of math and science.

Vector quantities

Main article: Vector quantity

In mathematics and physics, a vector is a special kind of number. It shows more than just size—it also shows direction. Vectors are useful because we can add them together and change them by multiplying with other numbers. This helps us solve many problems in science and engineering.

Vector spaces

Main article: Vector space

In mathematics and physics, a vector is a special kind of number that shows both size and direction. Vectors help us describe things like force or speed in a specific direction. A vector space is a group of these vectors. In this group, you can add vectors together or change their size by multiplying them by numbers, called scalars, and they will still stay in the same group.

Vectors in algebra

Every algebra over a field is a special kind of space. The pieces in this space are sometimes called vectors. This name comes from old ideas.

Some special kinds of vectors are:

• Vector quaternion, a type of number without a real part
• Multivector or p-vector, part of a bigger math idea called exterior algebra
• Spinors, also called spin vectors, used to understand turns and spins in space. They are 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 show many types of information. For example, a rotation vector tells us the direction and angle of a spin. A Burgers vector explains changes in crystal structures. In music, an interval vector keeps track of the distances between notes.

In statistics, a probability vector shows chances that always add up to one, and a random vector groups many changing values together. A logical vector is just 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.