Vector-Valued Function

Topics: Calculus

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(definition)

A vector-valued function $\text{r}(t)$ is a mapping from its domain $D\subset \mathbb{R}$ to its range $R\subset V_3$ (a vector space of dimension 3), so that for each $t$ in $D$, $\text{r}(t)=v$ for exactly one vector $v\in R$.

We can write a vector-valued function as:

$$\text{r}(t)=f(t)\text{i}+g(t)\text{j}+h(t)\text{k}$$

…for some scalar functions $f$, $g$ and $h$ (called the component functions of $r$).

Tip $\text{i, j, k}$ and the basis

Remember that, in this case, the basis of $V_3$ is $\{\mathbf{i},\mathbf{j},\mathbf{k}\}$. As such, we are writing $\text{r}(t)$ as a linear combination of the vectors in this basis.

Normal vs. Vector-Valued Function Graphs

Notice that a graph generated by a vector-valued function differs from the one generated by a normal function in that the former doesn’t include its domain, while the latter does. The graph of a vector-valued function is traced in its counterdomain.