Problem 100 - Calculus Volume 3

Question

True or False? Function \(\mathbf{r}(t)=\mathbf{a}+t(\mathbf{b}-\mathbf{a})\), where \(0 \leq t \leq 1\), parameterizes the straight-line segment from a to \(\mathbf{b}\).

Step-by-Step Solution

Answer

True, the function parameterizes the line segment from \( \mathbf{a} \) to \( \mathbf{b} \).

Step 1: Understand the Line Segment Parameterization

The expression \( \mathbf{r}(t) = \mathbf{a} + t(\mathbf{b} - \mathbf{a}) \) represents a linear interpolation between the points \( \mathbf{a} \) and \( \mathbf{b} \). When \( t = 0 \), \( \mathbf{r}(t) = \mathbf{a} \) and when \( t = 1 \), \( \mathbf{r}(t) = \mathbf{b} \). This suggests that \( \mathbf{r}(t) \) should trace the straight-line segment from \( \mathbf{a} \) to \( \mathbf{b} \) as \( t \) varies from 0 to 1.

Step 2: Analyze the Intermediate Values of t

For any \( 0 < t < 1 \), \( \mathbf{r}(t) = \mathbf{a} + t(\mathbf{b} - \mathbf{a}) \) results in a point that lies on the straight path from \( \mathbf{a} \) to \( \mathbf{b} \). This is because \( \mathbf{r}(t) \) is a linear combination of \( \mathbf{a} \) and \( \mathbf{b} \) weighted by \( 1-t \) and \( t \), respectively.

Step 3: Conclusion Based on Analysis

Since for \( t = 0 \) and \( t = 1 \), the function \( \mathbf{r}(t) \) passes exactly through \( \mathbf{a} \) and \( \mathbf{b} \), and for \( 0 < t < 1 \), it traces a straight line between these two endpoints, the parameterization is accurate. Therefore, the statement about the parameterization being a line segment from \( \mathbf{a} \) to \( \mathbf{b} \) is true.

Key Concepts

Linear Interpolation

Linear interpolation is a method used to construct new data points within the range of a set of known data points. It’s a very handy tool when you wish to find values between two endpoints, especially in geometry and computational graphics. In the context of our exercise, the formula \( \mathbf{r}(t) = \mathbf{a} + t(\mathbf{b} - \mathbf{a}) \) is an example of linear interpolation being used to parameterize a line segment between two points \( \mathbf{a} \) and \( \mathbf{b} \).

The parameter \( t \) allows us to control our position along the line segment.

• When \( t = 0 \), we are exactly at point \( \mathbf{a} \).
• When \( t = 1 \), we reach point \( \mathbf{b} \).
• For \( 0 < t < 1 \), we lie somewhere between \( \mathbf{a} \) and \( \mathbf{b} \), making this a true line segment parameterization as it constructs points directly along the path between the two endpoints.

This makes linear interpolation a simple yet powerful technique to create smooth transitions and scalable models in both mathematical and real-world problems.

Vector Functions

Vector functions are mathematical expressions that assign a vector to every real number in some interval. In many sciences and engineering disciplines, vector functions are used to describe a variety of physical phenomena. They provide the ability to describe movements and paths in multidimensional space.

The function \( \mathbf{r}(t) = \mathbf{a} + t(\mathbf{b} - \mathbf{a}) \) is a vector function that is linear in nature. Here, for each value of \( t \) in the interval \([0, 1]\), a unique vector is created. The vectors will point to different positions along the line segment from \( \mathbf{a} \) to \( \mathbf{b} \).

• Such representations are highly useful in computer graphics and physics to simulate actions, positions, and movements.
• They can resolve into velocity and acceleration vectors as derivatives.
• Vector functions provide a uniform way to handle complex mathematical tasks in multiple dimensions.

Understanding how a vector behaves through the function allows us a precise representation of lines, surfaces, and other geometric constructs.

Parameterization of Curves

Parameterization of curves is a way to represent a curve by expressing the coordinates of the points on the curve as functions of a variable, generally noted as \( t \). These functions are called parameterized equations.

For line segments, such as the one defined by \( \mathbf{r}(t) = \mathbf{a} + t(\mathbf{b} - \mathbf{a}) \), parameterization serves to describe exactly where you are on the curve based on the input \( t \). With \( t \) ranging from 0 to 1,

• The parameterization captures the entirety of the line segment from \( \mathbf{a} \) to \( \mathbf{b} \).
• This technique is ideal in computer graphics for rendering curves.
• It is used in physics to describe trajectories and in mathematics to explore new types of curves.

By using parameterization, one can easily manipulate and analyze the properties of the curve. Parameterization is especially useful for complex curves where straightforward algebraic descriptions are impractical.