A *vector* is the mathematical representation of an arrow. While many quantities like temperature are a scalar, i.e. a single number, other quantities like velocity are a vector: a car travels into a certain direction with a certain speed. The direction is given by the direction of the vector and the speed is encoded in its length. ![[vector.excalidraw.light.svg]] Not only velocity can be represented as a vector, but also [[Quantum State|quantum states]] and [[spin]] are represented by vectors. In a more mathematical sense, a vector is represented by a column of numbers: ![[vector_math.excalidraw.light.svg]] In this notation, the upper number determines the length in the $x$ direction and the lower number determines the length in the $y$ direction. In the plot above, the orange arrow ($\vec{z}$) is the addition of the two vectors $\vec{x}_1$ (red) and $\vec{x}_2$ (blue). In mathematics, vectors are organized in so-called **vector spaces**. These spaces host all vectors that can be expressed as multiples and additions of a set of vectors called [[Basis|basis]]. Certain vector spaces, so-called **Hilbert spaces** play an essential role in [[Quantum Mechanics|quantum mechanics]] since they build the framework to describe [[Quantum State|quantum states]]. >[!read]- Further Reading >- [[Quantum State]] >[!ref]- References