Part 4/6:
In this representation, ( f(t), g(t), h(t) ) are our component functions, each being a real-valued function that contributes to the overall vector output of ( \mathbf{R}(t) ). These components must correspond to the three axes in 3D space, allowing us to graph the vector function visually.
Example of a Vector Function
Let's consider a practical example to understand the domain of a vector function. Assume we have the vector function:
[
\mathbf{R}(t) = \begin{pmatrix} t \ t^3 \ \sqrt{3 - t} \end{pmatrix}
]
Here, ( f(t) = t ), ( g(t) = t^3 ), and ( h(t) = \sqrt{3 - t} ).
To identify the domain of ( \mathbf{R}(t) ), we need to ensure that all components are well-defined.
- ( t ) and ( t^3 ) are defined for all real numbers.