Question

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

Short answer

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

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.

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.

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.

Key concepts

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’s 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.