Euclidean vector

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.