Part 2/7:
Vector functions can be defined as functions that take real numbers as input and produce vectors as outputs. They extend our understanding of functions beyond the real-valued functions we are familiar with, enabling us to describe curves and surfaces in space.
The fundamental form of a vector function ( \mathbf{R}(t) ) in three-dimensional space can be expressed as:
[
\mathbf{R}(t) = \langle f(t), g(t), h(t) \rangle
]
where ( f(t), g(t), ) and ( h(t) ) are real-valued functions that correspond to the x, y, and z coordinates respectively. This definition allows us to visualize complex curves, such as in orbital mechanics or computer graphics, where objects travel along paths defined by vector functions.