Question

How many dependent scalar variables does the function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) have?

Short answer

Answer: The function has 3 dependent scalar variables.

Step-by-step solution

Identify the vector function given

The vector function is given as \(\mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle\), which means it takes an input variable t and outputs a vector with coordinates (or components) \(f(t)\), \(g(t)\), and \(h(t)\).

Identify the dependent scalar variables

From the vector function, we see that the scalar functions \(f(t)\), \(g(t)\), and \(h(t)\) are dependent on the input variable t. These scalar functions are the dependent scalar variables.

Count the dependent scalar variables

In the vector function, there are three dependent scalar variables: \(f(t), g(t)\), and \(h(t)\). So, the function \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) has 3 dependent scalar variables.

Key concepts

Dependent Scalar Variables

When we talk about dependent scalar variables in the context of a vector function, we are referring to the individual scalar components of a vector that change in response to another variable, known as the input variable. In the exercise \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\), \(f(t)\), \(g(t)\), and \(h(t)\) are dependent scalar variables because they depend on the value of \(t\).

Why is understanding dependent scalar variables important? It plays a crucial role in fields such as physics and engineering where vectors describe quantities such as velocity and force. Each scalar function in a vector provides valuable information about the vector's magnitude and direction at a given point in time. Recognizing these dependent variables allows us to analyze how a system evolves or responds to different conditions.

For instance, if our vector function represents the position of a particle over time, \(f(t)\), \(g(t)\), and \(h(t)\) would give us the particle's coordinates in a three-dimensional space at any time \(t\). Thus, the exercise tells us that there are three elements that are subject to change based on the input, making the analysis three-dimensional.

Input Variable

An input variable, often denoted by \(t\) in the context of vector functions, is essentially the independent variable upon which other values are based. It is analogous to the 'x' often seen in basic functions like \(y=f(x)\). However, instead of mapping to a single value, this input variable is mapped to an entire vector, which has both magnitude and direction.

In the expression \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\), \(t\) is the independent input variable. It can represent a multitude of things such as time, distance, or any other quantifiable entity that can vary independently. As \(t\) varies, it influences the dependent scalar variables—allowing us to study how each component changes. In this way, a vector function is a more complex and informative function than a standard scalar function.

Imagine \(t\) as a slider moving along a timeline; as it shifts, the components \(f(t)\), \(g(t)\), and \(h(t)\) adjust accordingly, dynamically shaping the vector's representation over the course of the input's range. Understanding the role of the input variable is essential for interpreting and visualizing the behavior of the vector function across different input values.

Components of a Vector

The components of a vector are the individual parts that make up the whole. In three-dimensional space, vectors are typically described using three components, which can be thought of as instructions on how to get from one point to another by moving along the x, y, and z-axis, respectively. The notation \(\mathbf{r}(t)=\langle f(t), g(t), h(t)\rangle\) from the exercise shows a vector with its components each being a function of \(t\).

Why is this significant? Well, in many scientific and mathematical applications, vectors are used to describe physical phenomena such as forces, velocities, and accelerations. Each component of a vector carries with it specific information about the direction and magnitude in relation to its respective axis. For example, \(f(t)\) might indicate movement forward or backward, \(g(t)\) up or down, and \(h(t)\) left or right.

Understanding the components of a vector not only helps in visualizing movements and changes within a physical space but also aids in performing vector operations such as addition, multiplication, and finding the magnitude and direction. Thus, having a solid grasp on what each component signifies is crucial for anyone aiming to work with vectors in any practical application.