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# 2.1: Vector Valued Functions

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## Definition of a Vector Valued Function

A vector valued function is a function where the domain is a subset of the real numbers and the range is a vector.

In two dimensions

$\textbf{r}(t)=x(t)\hat{\textbf{i}}+y(t) \hat{\textbf{j}}.$

In three dimensions

$\textbf{ r}(t)=x(t)\hat{\textbf{i}}+y(t) \hat{\textbf{j}}+ z(t) \hat{\textbf{k}}.$

You will notice the strong resemblance to parametric equations. In fact there is an equivalence between vector valued functions and parametric equations.

Example $$\PageIndex{1}$$

$\textbf{ r}(t)=3\hat{\textbf{i}}+t\hat{\textbf{j}} +(\sin t) \hat{\textbf{k}}$

To graph a vector valued function we can just graph the parametrically defined function.

Example $$\PageIndex{2}$$

Sketch the graph of

$\textbf{ r}(t)=(t-1)\hat{\textbf{i}}+t^2 \hat{\textbf{j}}$

Solution

We draw vectors for several values of t and connect the dots. Notice that the graph is the same as $$y=(x+1)^2$$. ## Limits

We define the limit of a vector valued function by taking the limit of each of the components. Formally

Definition: Limit of a Vector Valued Function

$\lim_{t\to{t_0}} \textbf{r}(t) = (\lim_{t\to{t_0}} x(t)) \hat{\textbf{i}} + (\lim_{t\to{t_0}} y(t)) \hat{\textbf{j}} + (\lim_{t\to{t_0}} z(t)) \hat{\textbf{k}}.$

Example $$\PageIndex{3}$$

Find the limit

$\lim_{t\to{0}} r(t)$

if

$\textbf{r}=e^t \hat{\textbf{i}} + \dfrac{\sin t}{t} \hat{\textbf{j}} + (t \ln t ) \hat{\textbf{k}}.$

Solution

We take the three limits one at a time

The first function is continuous at $$t = 0$$, so we can just plug in to get

$e^0 =1.$

For the second function, we get 0/0, so we use L'Hospital's rule to get

$\dfrac{\cos t}{1}.$

Now plug in to get

$\dfrac{1}{1}=1.$

For the $$k^{\text{th}}$$ component, we rewrite as

$\dfrac{\ln t}{\frac{1}{t}}.$

Now use L'Hospital's rule to get

$\dfrac{\frac {1}{t}}{-\frac{1}{t^2}}=t.$

Plugging in 0 gives 0. Finally, gathering our results gives a limit of

$\hat{\textbf{i}} + \hat{\textbf{j}}.$

## Continuity

We define continuity of vector valued functions in a similar way to how continuity of real valued functions was defined.

Definition: Continuous Functions

A vector valued function is continuous at $$t_0$$ if it is defined at $$t_0$$ and

$\lim_{t\to{t_0}} r(t)=r(t_0).$

The practical way to investigate continuity is to look at each of the components.

Example $$\PageIndex{4}$$

Determine where the following vector valued function is continuous.

$\textbf{r} (t) = \ln (1-t) \hat{\textbf{i}} + \dfrac{1}{t} \hat{\textbf{j}} + 3t \hat{\textbf{k}}.$

Solution

The first component is continuous for all values of $$t$$ less than 1, the second component is continuous for $$t$$ nonzero, and the third component is continuous for all real numbers. We can conclude that $$\textbf{r} (t)$$ is continuous for all $$t$$ less than 1 but not equal to 0.

Larry Green (Lake Tahoe Community College)

• Integrated by Justin Marshall.