Vector-valued functions and differentiation of vector

Vector valued functions and differentiation of vector

Vector-Valued Functions

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We are very familiar with real valued functions, that is, functions whose output is a real number. This section introduces vector-valued functions — functions whose output is a vector.

Vector-Valued Functions. 

 

A vector-valued function is a function of the form

$$\overrightarrow{r}\left(t\right)=⟨f\left(t\right),g\left(t\right)⟩\text{or}\overrightarrow{r}\left(t\right)=⟨f\left(t\right),g\left(t\right),h\left(t\right)⟩\text{,}$$

where $f\text{,}$ $g$ and $h$ are real valued functions.

The domain of $\overrightarrow{r}$ is the set of all values of $t$ for which $\overrightarrow{r}\left(t\right)$ is defined. The range of $\overrightarrow{r}$ is the set of all possible output vectors $\overrightarrow{r}\left(t\right)\text{.}$

 Evaluating and Graphing Vector-Valued Functions

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Evaluating a vector-valued function at a specific value of $t$ is straightforward; simply evaluate each component function at that value of $t\text{.}$ For instance, if $\overrightarrow{r}\left(t\right)=⟨t^2,t^2+t-1⟩\text{,}$ then $\overrightarrow{r}(-2)=⟨4,1⟩\text{.}$ We can sketch this vector, as is done in fig a . Plotting lots of vectors is cumbersome, though, so generally we do not sketch the whole vector but just the terminal point. The graph of a vector-valued function is the set of all terminal points of $\overrightarrow{r}\left(t\right)\text{,}$ where the initial point of each vector is always the origin. In fig b .  we sketch the graph of $\overrightarrow{r}$ ; we can indicate individual points on the graph with their respective vector, as shown.

fig(a)

Fig b

Vector-valued functions are closely related to parametric equations of graphs. While in both methods we plot points $(x\left(t\right),y\left(t\right))$ or $(x\left(t\right),y\left(t\right),z\left(t\right))$ to produce a graph, in the context of vector-valued functions each such point represents a vector. The implications of this will be more fully realized in the next section as we apply calculus ideas to these functions.

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