6. Vector Valued Functions

Now that we have some vector basics down, we will look at curves in 2D and 3D. Curves are the result of vector valued functions (also known as parametric curves), functions of a single independent variable (input) and multiple dependent variables (outputs). Here's the notation:

$\large \vec{r}(t)=\left<x(t),y(t),z(t)\right>$

Here, $\large t$ is the independent variable (sometimes called the parameter), and $\large x(t)$, $\large y(t)$, and $\large z(t)$ are the dependent variables. We package the three output values into a vector. The output for any value of $\large t$ is a vector (called $\large \vec{r}(t)$ in this example).

The domain for a vector valued function (called the parameter domain) is a set of values for the independent variable $\large t$: $t_{min}<t<t_{max}$.

Let's see how this definition leads to a curve in space:

We will often think of the parameter $\large t$ as time. In that context, the vector valued function is carving out a curve in space (the tip of the vector is carving out a curve in space) as time moves forward. Our vector valued functions have a natural direction associated with them, which we represent with arrows along the curve as in the video.

There are many curves that can be created this way, and we will explore some in class and in homework. But there are three main vector valued functions that will come up repeatedly in this class:

Lines

The equation of a line in space is:

$$\large \vec{r}(t)=t\vec{v}+\vec{r_0}\text{ }, -\infty < t < \infty $$

Here, $\large \vec{r_0}$ is a point in space that our line passes through (the green dot in the image below), and $\large \vec{v}$ is a direction vector (like a slope, the brown vector in the image below), a vector that points in the direction the line goes.

Often, instead of a direction vector, we have two points. In class, we will do an example of how we can find the equation of the line if we have two points.

Line segments

The vector valued function for a line segment connecting a point $\large P\left(x_1,y_1,z_1\right)$ to a point $\large Q\left(x_2,y_2,z_2\right)$ is:

$$\large \vec{r}(t)=\overrightarrow{PQ}t+\vec{P}, \text{ }0\le t \le 1$$

where $\large \overrightarrow{PQ}$ is the vector going from point P to the point Q (found via Q-P), and $\large\vec{P}$ is the vector going from the origin to the point P.

Notice, this equation is essentially the same as the equation of the line, but the parameter domain just ranges from 0 to 1 instead of from $\large -\infty$ to $\large \infty$.

Circles

The vector valued function for a circle of radius $\large R$ centered at the origin is

$\large \vec{r}(t)=\left&lt;R\cos(t),R\sin(t)\right&gt;,\text{ } 0\le t \le 2\pi$

This function starts at the point $\large \left(1,0\right)$ and moves counterclockwise.

This may be a good time to point out that multiple vector valued functions can represent the same curve! In the case of a circle $\large \left&lt;R\sin(t),R\cos(t)\right&gt;$ and $\large \left&lt;-R\cos(t),R\sin(t)\right&gt;$ also lead to a circle of radius $\large R$.

The image below is the vector valued function $\large \vec{r}(t)=\left&lt;\cos(t),\sin(t),1\right&gt;$ (so the circle is at height $\large z=1$ instead of on the $\large xy$-plane):

Graphs

Our traditional graphs $\large y=f(x)$ also represent curves, and we can write these as vector valued functions in a simple way. We just let the parameter $\large t$ be the independent variable $\large x$!

$\large \vec{r}(t)=\left&lt;t,f(t)\right&gt;$

The image below is a vector valued function version of $\large y=e^{-t/2}\sin(3t)$:

Tomorrow's class:

• Practice working with vector valued functions

Section Outcomes:

I can:

• Find a vector valued function representing a line, a line segment, a circle (or part of a circle), and a graph.
• Plot points on the curve represented by a vector valued function.

Other resources

For more on vector valued functions, see https://openstax.org/books/calculus-volume-3/pages/3-1-vector-valued-functions-and-space-curves