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 scaled to form a vector space. A vector is often shown 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 developed over more than 200 years, with many people helping to shape it. In 1835, Giusto Bellavitis introduced the concept of "equipollence," treating pairs of parallel line segments of the same length and direction as equal.

Later, William Rowan Hamilton introduced the term "vector" as part of his work on quaternions, which combine real numbers and vectors. Other mathematicians, such as Hermann Grassmann, also worked on similar systems. By the late 1800s, Josiah Willard 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 that has both magnitude (like how long or strong something is) and direction (which way it points). You can think of it like an arrow that shows both how far and in which direction something is going.

Vectors are important because 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 usually shown as arrows in drawings. The start of the arrow is called the origin, and the end is called the head. The length of the arrow shows the size of the vector, and the direction of the arrow shows the direction of the vector.

When we need to work with vectors in math, we can use numbers to describe them. 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 goes in each direction.

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

Vectors can be described using different sets of directions, and the way we describe them depends on the problem we are solving. No matter which way we choose, the basic properties of the vector stay the same.

Properties and operations

See also: Vector notation § Operations

A Euclidean vector, often just called a vector, is a geometric object that has both size (called magnitude) and direction. These vectors can be added together and stretched or shrunk (scaled) by numbers to form a vector space.

Vectors are useful in many areas of math, physics, and engineering because they help describe both the amount and direction of quantities like force or motion. For example, if you push a box north with a force of 10 newtons, that push can be represented as a vector. The vector would have a magnitude of 10 (the strength of the push) and a direction of north (where the push is aimed).

Physics

Main article: Vector quantity

Vectors are very 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 or stretch our view. For example, if you turn around, the way we describe a moving object's speed stays linked to how we 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 a little 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 actually still spinning the same way. Other examples include magnetic fields and the twisty force on objects.