Euclidean vector

In mathematics, physics, and engineering, a Euclidean vector or simply a vector is a geometric object that has magnitude (or length) and direction. Vectors can be added and changed in size to form a vector space. A vector is often drawn as an arrow pointing from one place to another, showing both how far and in which direction you need to go.

The word vector comes from Latin and means 'carrier'. It was first used by 18th century astronomers studying how planets move around the Sun. The magnitude of a vector is the distance between two points, and the direction is the way you move from one point to the other.

Vectors are very important in physics. Things like the velocity and acceleration of a moving object, as well as the forces acting on it, can all be described using vectors. Even though these don't always show actual distances, they still have a magnitude and direction that can be shown with arrows. How we mathematically represent a vector depends on the coordinate system we use.

History

The idea of a vector took more than 200 years to develop, with help from many people. In 1835, Giusto Bellavitis introduced the idea of "equipollence," saying that parallel line segments of the same length and direction are equal.

Later, William Rowan Hamilton introduced the term "vector" in his work on quaternions, which mix real numbers and vectors. Other mathematicians, like Hermann Grassmann, also worked on similar ideas. By the late 1800s, JosiahWillard Gibbs and Edwin Bidwell Wilson helped create the modern system of vector analysis we use today.

Overview

In physics and engineering, a vector is a special kind of object. It has two parts: magnitude (how long or strong something is) and direction (which way it points). You can think of it like an arrow showing both how far and which way something is going.

Vectors are important. They help describe things like how fast something is moving (velocity) or how hard a push or pull is (force). These things need both a number for how much and a direction to be fully understood. Vectors make it easier to work with these ideas in math and science.

Representations

Further information: Vector representation

Vectors are often drawn as arrows. The place where the arrow starts is called the origin, and the point where it ends is called the head. The length of the arrow shows the size of the vector, and the direction of the arrow shows which way the vector is pointing.

In math, we can use numbers to describe vectors. For example, a vector going from the point (0, 0) to the point (2, 3) can be written as (2, 3). This tells us how far the vector moves in each direction.

In three dimensions, a vector can be written with three numbers, like (a₁, a₂, a₃). These numbers show the vector’s direction and size in three-dimensional space.

Vectors can be described using different sets of directions, depending on the problem we are solving. No matter which way we choose, the basic properties of the vector do not change.

Properties and operations

See also: Vector notation § Operations

A Euclidean vector, often just called a vector, is an object that has both size and direction. These vectors can be added together and made bigger or smaller by numbers.

Vectors are useful in math, physics, and engineering because they help describe how much of something there is and which way it is going. For example, if you push a box north with a force of 10 newtons, that push can be shown as a vector. The vector would have a size of 10 and a direction of north.

Physics

Main article: Vector quantity

Vectors are useful in physics and science. They help describe things that have both a size and a direction.

Length and units

In math, the size of a vector can change depending on what it represents. For example, if a vector shows a force, its size depends on units of force. Vectors of the same type usually have consistent sizes, but different types can look different even if they are the same length.

Vector-valued functions

In physics and math, vectors can change over time. For example, the position of a moving object can be shown as a vector that changes with time. We can find how fast the vector changes by looking at each part of it separately.

Position, velocity and acceleration

The position of a point in space can be shown as a vector from the starting point. The difference between two positions is a vector that shows how far and in which direction you need to move from one point to the other.

The velocity of a moving object is a vector. Its size shows how fast the object is moving. Acceleration is also a vector, showing how quickly the speed or direction of motion is changing.

Force, energy, work

Force is a vector that shows how much push or pull there is and in which direction. Work is done when a force moves an object from one place to another.

Vectors, pseudovectors, and transformations

Vectors can change when we look at them from different angles. For example, if you turn around, how we describe a moving object's speed still relates to how you turned. This special kind of vector is called a contravariant vector. Things like how far something moves, how fast it goes, or the push it feels are all contravariant vectors.

Some vectors act differently when we flip our view, like in a mirror. These are called pseudovectors. One example is how fast a wheel spins when you drive a car. If you look in a mirror, it might seem to spin the other way, but it’s still spinning the same way. Other examples include magnetic fields and the twisty force on objects.