In the context of calculus (as well as in physics and engineering), a vector is a geometry object that has both magnitude and direction. This concept is also known as “Euclidean vector” or “geometric vector”. Vectors are fundamental to the language of classical physics: velocity, acceleration, and force are all vectors.
In the much more general context of linear algebra, a vector can be any object in a vector space which can be manipulated according to the rules of vector algebra.
In the two-dimensional plane, a vector can be represented by two numbers — the vertical and horizontal “components”. They form the numerical coordinates of a vector in standard basis.
\[ \vec{v} = x \vec{x} + y \vec{y} \]
\[ \begin{aligned} &\|\vec{v}\| &&\text{the magnitude of } \vec{v} \\ &\vec{v} + \vec{u} &&\text{the sum of two vectors} \\ &-\vec{v} &&\text{reverse the direction (and keeping the magnitude)} \\ &c \cdot \vec{v} &&\text{scaling by a (positive) factor of } c \end{aligned} \]