Euclidean vector

An arrow can show a way to move.

It shows how far to go. This is the length. It also shows which way to go. This is the direction.

We use these arrows in science. They can show a push or a pull. This is called a force. They can also show speed.

Some arrows start at one spot. They end at another spot.

Other arrows do not need a spot. They only care about length and way.

We can even add arrows together.

This helps us see a new path. It is a very helpful tool.

Imagine you want to move from one spot to another.

You need to know how far to go. You also need to know which way to head. In math, we use a tool called a vector to show this. A vector is like an arrow. It has a length, which we call magnitude. It also has a direction.

Scientists use vectors to describe the world. They use them to show force, which is a push or a pull. They also use them for velocity. Velocity is how fast something moves in a certain direction.

Vectors can be tied to a specific spot. We call these bound vectors. Other vectors do not need a fixed starting point. We call these free vectors. For a free vector, only the length and direction matter.

Many smart people helped create this idea over 200 years. William Rowan Hamilton used the word vector. It comes from a Latin word that means "to carry." Later, Josiah Willard Gibbs helped make it a tool for engineers. This helped people work with shapes in three dimensions.

Imagine you are trying to describe a movement to a friend. You cannot just say you moved five meters. You must also say which way you went. In math, we use a special tool called a vector to show this.

A vector is a geometric object that has two main parts. The first part is the magnitude, which is just a fancy word for length. The second part is the direction. We often draw a vector as an arrow. The arrow starts at an initial point and ends at a terminal point.

This arrow shows exactly how to "carry" a point from one place to another.

Vectors work in many different ways through math. You can add two vectors together to find a new path. You can also subtract them or multiply them by a number.

When you multiply a vector by a number, you can change its size. This is called scaling. For example, multiplying a vector by three makes it three times longer.

These operations follow rules similar to regular numbers. You can use these rules to solve hard problems in physics. Engineers use these steps to understand how things move in three-dimensional space.

The idea of the vector grew slowly over two hundred years. It was not invented by just one person. In 1835, a mathematician named Giusto Bellavitis helped start the idea. He looked at parallel lines of the same length. Later, William Rowan Hamilton used the actual word "vector." The word comes from a Latin term that means "to carry."

Hamilton used it as part of a larger system called quaternions. Other thinkers like Hermann Grassmann also made important discoveries. Grassmann wrote about these ideas in 1840, but many people ignored him at first.

Many famous scientists helped shape the modern system we use today. Peter Guthrie Tait worked with Hamilton's ideas in 1867. Then, William Kingdon Clifford published work in 1878. Clifford made math easier for engineers by splitting certain parts of the math apart.

Later, Josiah Willard Gibbs created a very helpful system. He published his work in 1881. His method became the standard way to study vectors. In 1901, Edwin Bidwell Wilson published a book that helped spread these ideas even further. These many steps turned a complex idea into a useful tool.

We see vectors in action all around us every day. Scientists use them to describe velocity, which is speed with a direction. They also use them to describe force, like a push or a pull.

Even acceleration can be shown with a vector. If you are pushing a heavy box, that push is a vector. If a car is driving north at a certain speed, that is also a vector. Without vectors, it would be very hard to map out how objects move through our world. They help us turn physical actions into clear, mathematical pictures.

A Euclidean vector is a geometric object defined by two essential properties: magnitude and direction. In many contexts, such as physics and engineering, a vector is represented as a directed line segment, or an arrow.

The arrow begins at an initial point, labeled A, and ends at a terminal point, labeled B. The magnitude of the vector is the distance between these two points. The direction refers to the specific displacement from the starting point to the ending point. While many people think of vectors only as arrows in space, pure mathematicians define them more broadly. In that field, a vector is any element belonging to a vector space. This abstract definition allows for vectors that may not have a physical length or a visual direction.

Vectors can be classified into different types based on how they are used. A bound vector is a vector that has a definite, fixed initial and terminal point. This is often necessary in mechanics, where a force must be applied to a specific point of contact on an object. In contrast, a free vector is defined only by its magnitude and direction. For a free vector, the specific starting and ending points do not matter. Two arrows represent the same free vector if they are equipollent, meaning they have the same length and point the same way.

If you place two such arrows in a way that they form a parallelogram, they are considered equivalent. In a coordinate system with an origin, a free vector can be treated as a bound vector starting at that origin. This allows mathematicians to represent vectors using numerical coordinates.

Algebraic operations allow us to manipulate vectors in ways that mirror regular arithmetic. We can perform addition, subtraction, multiplication, and negation on vectors. These operations follow specific mathematical laws, such as commutativity, associativity, and distributivity. For example, adding two vectors can be visualized as following one path after another.

We can also use scalar multiplication to change the size of a vector. When you multiply a vector by a number, you are scaling it.

If the number is greater than one, the vector stretches. If the number is between zero and one, the vector shrinks. Multiplying by a negative number will also change the direction of the vector. These operations make vectors a powerful tool for solving complex spatial problems.

The concept of the vector developed gradually over more than 200 years. It was not the work of a single person but a result of many contributions. In 1835, Giusto Bellavitis established the concept of equipollence. He worked in a Euclidean plane and realized that parallel line segments with the same length and orientation could be treated as equivalent. This was one of the first steps toward creating a vector space. Later, William Rowan Hamilton introduced the term "vector." The word comes from the Latin "vehere," which means "to carry." Hamilton used vectors as part of a larger system called quaternions, where the vector was the imaginary part.

Throughout the nineteenth century, several other mathematicians expanded these ideas. Hermann Grassmann published a work in 1840 called "Theory of the Ebb and Flow." This was the first system of spatial analysis similar to modern vector systems. It included ideas that we now recognize as the cross product and the scalar product. Although his work was largely ignored until the 1870s, it was foundational. Other contributors included Augustin Cauchy, August Möbius, and Peter Guthrie Tait. Tait worked to carry forward Hamilton's quaternion standards in his 1867 treatise.

These researchers were building the framework that would eventually allow engineers to work in three dimensions.

In 1878, William Kingdon Clifford helped simplify these complex mathematical systems. He isolated the dot product and the cross product from the full quaternion product. This simplification was crucial because it made vector calculations accessible to engineers. Before this, the math was often too difficult for practical application. Later, Josiah Willard Gibbs further refined the field. He separated the vector part of quaternions to create an independent system. His work, published in 1881, is essentially the modern system of vector analysis used today. In 1901, Edwin Bidwell Wilson published a textbook that helped cement this modern approach by removing mentions of quaternions.

Vectors are essential for describing the physical world. In physics, many quantities are vector-valued, meaning they require both a number and a direction. Velocity is a primary example; it describes both the speed of an object and the direction it is moving.

Force is another example, as a push or a pull has a specific strength and a specific direction. Acceleration and momentum are also described using vectors. Even complex phenomena, such as electric and magnetic fields, are represented as vector fields. However, not all physical quantities are vectors. For instance, angular displacement and electric current do not follow the rules of vector addition, so they are not considered vectors. By using vectors, scientists can turn physical movements and forces into precise mathematical models.