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Chapter 11: Problem 3

Why is \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) called a vector-valued function?

Short Answer

Expert verified
Answer: Yes, \(\textbf{r}(t)\) is a vector-valued function because it takes a scalar input \(t\) and returns a vector with three scalar components, \(f(t)\), \(g(t)\), and \(h(t)\).

Step by step solution

01

Check the form of the function

Let's start by writing down the given function again: $$\mathbf{r}(t) = \langle f(t), g(t), h(t)\rangle$$ Notice that the function \(\mathbf{r}(t)\) takes in a scalar input, namely \(t\), and returns a vector with three components, \((f(t), g(t), h(t))\).
02

Examine the components

Now let's examine the components of the vector individually: 1. \(f(t)\): This is a function that takes the scalar input \(t\) and returns a scalar output. 2. \(g(t)\): This is another function that takes the scalar input \(t\) and returns a scalar output. 3. \(h(t)\): This is yet another function that takes the scalar input \(t\) and returns a scalar output. These components together form a vector \(\textbf{r}(t) = \langle f(t), g(t), h(t)\rangle\) for each input value \(t\).
03

Conclusion

Since \(\textbf{r}(t)\) takes in a scalar input \(t\) and returns a vector (with three scalar components, \(f(t)\), \(g(t)\), and \(h(t)\)), we can conclude that \(\textbf{r}(t)\) is indeed a vector-valued function.

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Most popular questions from this chapter

Let \(\mathbf{u}(t)=\left\langle 1, t, t^{2}\right\rangle, \mathbf{v}(t)=\left\langle t^{2},-2 t, 1\right\rangle\) and \(g(t)=2 \sqrt{t}\). Compute the derivatives of the following functions. $$\mathbf{u}(t) \cdot \mathbf{v}(t)$$

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\left\langle t^{4}-3 t, 2 t-1,10\right\rangle$$

Compute the following derivatives. $$\frac{d}{d t}\left(\left(t^{3} \mathbf{i}-2 t \mathbf{j}-2 \mathbf{k}\right) \times\left(t \mathbf{i}-t^{2} \mathbf{j}-t^{3} \mathbf{k}\right)\right)$$

Properties of dot products Let \(\mathbf{u}=\left\langle u_{1}, u_{2}, u_{3}\right\rangle\) \(\mathbf{v}=\left\langle v_{1}, v_{2}, v_{3}\right\rangle,\) and \(\mathbf{w}=\left\langle w_{1}, w_{2}, w_{3}\right\rangle .\) Prove the following vector properties, where \(c\) is a scalar. $$\mathbf{u} \cdot(\mathbf{v}+\mathbf{w})=\mathbf{u} \cdot \mathbf{v}+\mathbf{u} \cdot \mathbf{w}$$

Find the function \(\mathbf{r}\) that satisfies the given conditions. $$\mathbf{r}^{\prime}(t)=\left\langle 1,2 t, 3 t^{2}\right\rangle ; \mathbf{r}(1)=\langle 4,3,-5\rangle$$
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