Goal: Graph of a vector-valued function
Considerations
A vector is defined by two quantities: magnitude and direction. If we consider an initial point and move in a certain direction according to a certain distance, we arrive at a second point. The second point is the terminal point. To determine the coordinates of a vector, we subtract the coordinates of the original point from those of the terminal point.
A vector is said to be in standard position if if its original point is located in the origin. In order to maintain the uniqueness of the graph of a vector-valued function, we start by graphing the vectors in the domain of the function.. The graph of a vector-valued function of the form r(t) = f(t)i + g(t)j is the sets of all (t, r(t)) and the path it traces is called a plane curve. The graph of a vector-valued function of the form r(t) = f(t)i + g(t)j + h(t)k and the path it traces is called a space curve. Any representation of a plane curve or space curve using a vector-valued function is called a vector parameterization of the curve.
Examples
Create a graph of each of the vector-valued functions:
a. The plane curve represented by r(t) = 4costi + 3sintj 0 ≤ t ≤ 2ℼ
b. The plane curve represented by r(t) = 4costi + 3sintj 0 ≤ t ≤ π
Solutions
a. We start by creating a table of values made of values of t in the domain of the vector valued-function and corresponding values of r(t). We graph each of the vectors of the second column in standard form and connect the terminal point of each vector to form a curve. It turns out that the curve is an ellipse centered at the origin
Graph
b. Let's do the same for the same for the second example.
Table of values
Graph
The graph of this curve is an also an ellipse centered at the origin.
Practice
Create of the vector-valued function:
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