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Vector Function1

# Vector Function1 - Vector Functions We first saw vector...

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Vector Functions We first saw vector functions back when we were looking at the Equation of Lines . In that section we talked about them because we wrote down the equation of a line in in terms of a vector function (sometimes called a vector-valued function ). In this section we want to look a little closer at them and we also want to look at some vector functions in other than lines. A vector function is a function that takes one or more variables and returns a vector. We’ll spend most of this section looking at vector functions of a single variable as most of the places where vector functions show up here will be vector functions of single variables. We will however briefly look at vector functions of two variables at the end of this section. A vector functions of a single variable in and have the form, respectively, where , and are called the component functions . The main idea that we want to discuss in this section is that of graphing and identifying the graph given by a vector function. Before we do that however, we should talk briefly about the domain of a vector function. The domain of a vector function is the set of all t ’s for which all the component functions are defined. Example 1 Determine the domain of the following function. Solution The first component is defined for all t ’s. The second component is only defined for . The third component is only defined for . Putting all of these together gives the following domain. This is the largest possible interval for which all three components are defined.

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Let’s now move into looking at the graph of vector functions. In order to graph a vector function all we do is think of the vector returned by the vector function as a position vector for points on the graph. Recall that a position vector, say , is a vector that starts at the origin and ends at the point . So, in order to sketch the graph of a vector function all we need to do is plug in some values of t and then plot points that correspond to the resulting position vector we get out of the vector function. Because it is a little easier to visualize things we’ll start off by looking at graphs of vector functions in . Example 2 Sketch the graph of each of the following vector functions. (a) [ Solution ] (b) [ Solution ] Solution (a) Okay, the first thing that we need to do is plug in a few values of t and get some position vectors. Here are a few, So, what this tells us is that the following points are all on the graph of this vector function. Here is a sketch of this vector function.
In this sketch we’ve included many more evaluations that just those above. Also note that we’ve put in the position vectors (in gray and dashed) so you can see how all this is working. Note however, that in practice the position vectors are generally not included in the sketch. In this case it looks like we’ve got the graph of the line

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Vector Function1 - Vector Functions We first saw vector...

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