A gradient vector is a mathematical concept used in calculus to describe the direction and rate of change of a function. It consists of partial derivatives, which indicate how the function changes with respect to each variable. The gradient points in the direction of the steepest ascent, helping to identify where a function increases most rapidly. In a three-dimensional space, the gradient vector is represented as a three-dimensional arrow. Each component of the vector corresponds to the rate of change in the respective direction, making it useful in fields like physics, engineering, and machine learning for optimization and analysis of functions.