Part 3/6:
Vector functions are pivotal for modeling various phenomena in physics and engineering, particularly in describing curves and surfaces in space. For example, they serve as tools for analyzing the motion of objects, including planets revolving around the sun. A key application of vector functions is the derivation of Kepler's laws of planetary motion, which we will discuss more in-depth in subsequent sections.
Kepler's first law states that planets orbit the sun in elliptical paths, highlighting the importance of understanding the geometric properties the vectors can represent.
Understanding the Components
A vector function takes a general form:
[
\mathbf{R}(t) = \begin{pmatrix} x(t) \ y(t) \ z(t) \end{pmatrix} = f(t) \mathbf{i} + g(t) \mathbf{j} + h(t) \mathbf{k}
]