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Chapter 12: Problem 3

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

Short Answer

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**Question:** Explain why the function \(r(t) = \langle f(t), g(t), h(t) \rangle\) is called a vector-valued function. **Answer:** The function \(r(t) = \langle f(t), g(t), h(t) \rangle\) is called a vector-valued function because it takes a scalar input, \(t\), and produces a vector as its output. The output vector belongs to the three-dimensional real vector space \(\mathbb{R}^3\) and can be written in component form as \(\begin{bmatrix} f(t) \\ g(t) \\ h(t) \end{bmatrix}\). The presence of the vector output and its belonging to a vector space are the key properties that make this function a vector-valued function.

Step by step solution

01

Definition of a vector-valued function

A vector-valued function is a function that takes a scalar input (in this case, \(t\)) and produces a vector as its output. The output vector usually belongs to a vector space, such as \(\mathbb{R}^n\), where \(n\) is the number of dimensions.
02

Components of the given function

In the given function, \(\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\), we have three component functions: \(f(t)\), \(g(t)\), and \(h(t)\). These functions map the scalar input \(t\) to scalar outputs \(f(t), g(t), h(t)\), respectively.
03

Output of the given function

When we substitute a value of \(t\) into the function, we obtain a vector that can be written in component form: $$ \mathbf{r}(t) = \begin{bmatrix} f(t) \\ g(t) \\ h(t) \end{bmatrix} $$ This output vector belongs to the three-dimensional real vector space \(\mathbb{R}^3\). Since the output of the function is a vector, this makes \(\mathbf{r}(t)\) a vector-valued function.

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Most popular questions from this chapter

The points \(O(0,0,0), P(1,4,6),\) and \(Q(2,4,3)\) lie at three vertices of a parallelogram. Find all possible locations of the fourth vertex.

Assume that \(\mathbf{u}, \mathbf{v},\) and \(\mathbf{w}\) are vectors in \(\mathrm{R}^{3}\) that form the sides of a triangle (see figure). Use the following steps to prove that the medians intersect at a point that divides each median in a 2: 1 ratio. The proof does not use a coordinate system. a. Show that \(\mathbf{u}+\mathbf{v}+\mathbf{w}=\mathbf{0}\) b. Let \(\mathbf{M}_{1}\) be the median vector from the midpoint of \(\mathbf{u}\) to the opposite vertex. Define \(\mathbf{M}_{2}\) and \(\mathbf{M}_{3}\) similarly. Using the geometry of vector addition show that \(\mathbf{M}_{1}=\mathbf{u} / 2+\mathbf{v} .\) Find analogous expressions for \(\mathbf{M}_{2}\) and \(\mathbf{M}_{3}\) c. Let \(a, b,\) and \(c\) be the vectors from \(O\) to the points one-third of the way along \(\mathbf{M}_{1}, \mathbf{M}_{2},\) and \(\mathbf{M}_{3},\) respectively. Show that \(\mathbf{a}=\mathbf{b}=\mathbf{c}=(\mathbf{u}-\mathbf{w}) / 3\) d. Conclude that the medians intersect at a point that divides each median in a 2: 1 ratio.

The definition \(\mathbf{u} \cdot \mathbf{v}=|\mathbf{u}||\mathbf{v}| \cos \theta\) implies that \(|\mathbf{u} \cdot \mathbf{v}| \leq|\mathbf{u}||\mathbf{v}|(\text {because}|\cos \theta| \leq 1) .\) This inequality, known as the Cauchy-Schwarz Inequality, holds in any number of dimensions and has many consequences. Use the vectors \(\mathbf{u}=\langle\sqrt{a}, \sqrt{b}\rangle\) and \(\mathbf{v}=\langle\sqrt{b}, \sqrt{a}\rangle\) to show that \(\sqrt{a b} \leq(a+b) / 2,\) where \(a \geq 0\) and \(b \geq 0\).

The points \(P, Q, R,\) and \(S,\) joined by the vectors \(\mathbf{u}, \mathbf{v}, \mathbf{w},\) and \(\mathbf{x},\) are the vertices of a quadrilateral in \(\mathrm{R}^{3}\). The four points needn't lie in \(a\) plane (see figure). Use the following steps to prove that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram. The proof does not use a coordinate system. a. Use vector addition to show that \(\mathbf{u}+\mathbf{v}=\mathbf{w}+\mathbf{x}\) b. Let \(m\) be the vector that joins the midpoints of \(P Q\) and \(Q R\) Show that \(\mathbf{m}=(\mathbf{u}+\mathbf{v}) / 2\) c. Let n be the vector that joins the midpoints of \(P S\) and \(S R\). Show that \(\mathbf{n}=(\mathbf{x}+\mathbf{w}) / 2\) d. Combine parts (a), (b), and (c) to conclude that \(\mathbf{m}=\mathbf{n}\) e. Explain why part (d) implies that the line segments joining the midpoints of the sides of the quadrilateral form a parallelogram.

Determine the equation of the line that is perpendicular to the lines \(\mathbf{r}(t)=\langle-2+3 t, 2 t, 3 t\rangle\) and \(\mathbf{R}(s)=\langle-6+s,-8+2 s,-12+3 s\rangle\) and passes through the point of intersection of the lines \(\mathbf{r}\) and \(\mathbf{R}\).
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