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Chapter 14: 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 \(\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\) can be called a vector-valued function. Answer: The function \(\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\) can be called a vector-valued function because it takes a scalar input \(t\) and results in a vector output with components determined by the scalar functions \(f(t)\), \(g(t)\), and \(h(t)\). These component functions represent the x, y, and z coordinates of the output vector, which makes \(\mathbf{r}(t)\) a vector-valued function in three-dimensional space.

Step by step solution

01

Define vector-valued functions

A vector-valued function is a function that takes one or more scalar inputs and gives a vector as its output. It can be represented in different forms such as using component functions and combining them to form a vector. The general form of a vector-valued function in three-dimensional space can be represented as \(\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\), where \(f(t)\), \(g(t)\), and \(h(t)\) are scalar functions.
02

Describe the scalar component functions

In the vector-valued function \(\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\), the scalar component functions represent the directional components of the output vector in the x, y, and z axes respectively. The function \(f(t)\) represents the component in the x-direction, \(g(t)\) represents the component in the y-direction, and \(h(t)\) represents the component in the z-direction.
03

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

Based on the definition of a vector-valued function and the properties of \(\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\), we can conclude that \(\mathbf{r}(t)\) is a vector-valued function because it takes a scalar input \(t\) and results in a vector output with components determined by the scalar functions \(f(t)\), \(g(t)\), and \(h(t)\). These component functions represent the x, y, and z coordinates of the output vector, which makes \(\mathbf{r}(t)\) a vector-valued function in three-dimensional space.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Scalar Functions
Scalar functions are fundamental in understanding how vector-valued functions work. These functions, like \( f(t), g(t), \text{and } h(t) \), are called 'scalar' because they take a single variable input, often time \( t \), and return a single real number. These outputs reflect different aspects of a problem.
In the context of vector-valued functions, each scalar function represents a direction in space. For example:
  • \( f(t) \) could describe movement along the x-axis.
  • \( g(t) \) could describe movement along the y-axis.
  • \( h(t) \) could describe movement along the z-axis.
These directional components combine to form a vector. So, even though each function works independently, together they create a comprehensive representation of movement or change across three dimensions.
Three-Dimensional Space
Three-dimensional space is where our vector-valued function lives and operates. It鈥檚 essential for describing real-world physics or engineering problems. Think of it as the room where three directions meet: length, width, and height.
When dealing with vector-valued functions:
  • The vector \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \) moves through this 3D space as \( t \) changes, tracing out a path or curve.
  • This movement can explain various phenomena, like the trajectory of a ball or the flow of water.
Understanding these three dimensions helps us visualize the effect each scalar function has within a framework where everything is interconnected, allowing for a fuller picture of the scenario being studied.
Component Functions
Component functions are the lifeblood of vector-valued functions. Each component function, \( f(t), g(t), \) and \( h(t) \), plays a crucial role in defining movement in three-dimensional space. They essentially "build" the vector by determining its direction and magnitude.
Here's how they work together:
  • \( f(t) \) determines how far along the x-axis the vector reaches.
  • \( g(t) \) does the same for the y-axis.
  • \( h(t) \) dictates the reach along the z-axis.
By combining these component functions, the vector \( \mathbf{r}(t) \) is formed, tailored to show exactly what happens at any given point in space as time changes. This capacity to represent dynamic processes makes vector-valued functions indispensable in fields like physics and engineering.

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Most popular questions from this chapter

Compute the indefinite integral of the following functions. $$\mathbf{r}(t)=\left\langle 5 t^{-4}-t^{2}, t^{6}-4 t^{3}, \frac{2}{t}\right\rangle$$

Three-dimensional motion Consider the motion of the following objects. Assume the \(x\) -axis points east, the \(y\) -axis points north, the positive z-axis is vertical and opposite g. the ground is horizontal, and only the gravitational force acts on the object unless otherwise stated. a. Find the velocity and position vectors, for \(t \geq 0\). b. Make a sketch of the trajectory. c. Determine the time of flight and range of the object. d. Determine the maximum height of the object. A golf ball is hit east down a fairway with an initial velocity of \(\langle 50,0,30\rangle \mathrm{m} / \mathrm{s} .\) A crosswind blowing to the south produces an acceleration of the ball of \(-0.8 \mathrm{m} / \mathrm{s}^{2}\).

Relationship between \(\mathbf{r}\) and \(\mathbf{r}^{\prime}.\) Consider the helix \(\mathbf{r}(t)=\langle\cos t, \sin t, t\rangle,\) for \(-\infty

Prove that \(r\) describes a curve that lies on the surface of a sphere centered at the origin \(\left(x^{2}+y^{2}+z^{2}=a^{2}\right.\) with \(a \geq 0\) ) if and only if \(\mathbf{r}\) and \(\mathbf{r}^{\prime}\) are orthogonal at all points of the curve.

Zero curvature Prove that the curve $$\mathbf{r}(t)=\left\langle a+b t^{p}, c+d t^{p}, e+f t^{p}\right\rangle$$ where \(a, b, c, d, e,\) and \(f\) are real numbers and \(p\) is a positive integer, has zero curvature. Give an explanation.
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