Vector operations involve mathematical processes that manipulate vectors, which are quantities defined by both magnitude and direction. Common operations include addition, subtraction, and scalar multiplication. When adding or subtracting vectors, their corresponding components are combined, while scalar multiplication involves multiplying a vector by a number, changing its magnitude but not its direction. Another important vector operation is the dot product, which calculates a single number from two vectors, indicating their directional relationship. The cross product, on the other hand, produces a new vector that is perpendicular to the plane formed by the two original vectors. These operations are fundamental in fields like physics and engineering, where vectors represent forces, velocities, and other directional quantities.