How to Graph Vector Functions

Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

Vector functions are often used to describe the position of an object in space. Graphing a vector function is one way to create a visual representation of the motion of the object. In this lesson, you'll learn the steps you need to take to graph vector functions.

What Is a Vector Function?

John stands at the free throw line and throws a basketball toward the goal. Will it make it in the goal? Well, that depends on a couple of factors, such as how fast he throws the ball and the angle he throws it at.

In fact, John can predict exactly where the ball will be at any time after he throws it just by knowing these two things. Since the ball is moving in both the horizontal and vertical directions at the same time, its position can be described using a vector, which is a quantity with a magnitude and a direction.

One way that a vector quantity, like the position of the basketball, can be represented is by giving the coordinates of the ball in space. For example, if the ball is currently three meters horizontally in front of John and 1.5 meters above the height at which he shot it, we could represent its position like this as a vector written like <3, 1.5> m.

The position of the basketball at any point in time can be represented by a vector.
basketball vector

A mathematical function that outputs a vector is known as a vector function. A vector function might look something like this: r(t) = <t, 2t-3>. This means that at t = 1, this function would give the vector <1, -1>, and at t = 2, this function would give the vector <2, 1>.

Making a Data Table for Vector Functions

It's often helpful to graph vector functions on an x-y coordinate plane. The first step in creating a graph of a vector function is to make a data table. Pick several values of the input variable and then use the function to determine the x and y coordinates of the vector that results. Let's look at how to do that for the vector function r(t) = <t, 2t-3>. A data table for this vector function would look something like this:

vector data table

Graphing Vector Functions

Once you have a data table, you can plot the x and y coordinates of the vector function. For our function given above, this would look like:

graph of vector function

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