Problem 26 - College Trigonometry

Question

In Exercises \(25-39\), find a parametric description for the given oriented curve. the directed line segment from \((-2,-1)\) to \((3,-4)\)

Step-by-Step Solution

Answer

The parametric equations are \( x(t) = -2 + 5t \) and \( y(t) = -1 - 3t \) for \( 0 \leq t \leq 1 \).

Step 1: Define Parameters

To describe a line segment using parametric equations, we introduce a parameter, usually denoted as \( t \). For a line segment, \( t \) will vary from 0 to 1.

Step 2: Determine Initial and Final Points

Identify the starting point and the terminal point of the line segment. Here, the starting point \( P_1 \) is \((-2, -1)\) and the terminal point \( P_2 \) is \((3, -4)\).

Step 3: Derive Direction Vector

Calculate the direction vector \( \mathbf{d} \) by subtracting the initial point from the terminal point: \[ \mathbf{d} = (3 - (-2), -4 - (-1)) = (5, -3) \].

Step 4: Write Parametric Equations

The parametric equations of the line segment are derived using the formula:\[ \mathbf{r}(t) = P_1 + t \mathbf{d} \] Substitute the values of \( P_1 \) and \( \mathbf{d} \): \[ \mathbf{r}(t) = (-2, -1) + t(5, -3) \]. Separate into component equations: \( x(t) = -2 + 5t \) and \( y(t) = -1 - 3t \).

Step 5: Write Final Parametric Description

Thus, the parametric description of the line segment is: \[ x(t) = -2 + 5t \] \[ y(t) = -1 - 3t \] for \( 0 \leq t \leq 1 \).

Key Concepts

Direction Vector

When working with parametric equations for a line segment, one crucial element is the direction vector. This vector essentially tells us what direction the line segment extends from the starting point to the ending point. To find the direction vector for a line segment, subtract the coordinates of the starting point from those of the ending point.
For example, if we have a starting point \( P_1 = (-2, -1) \) and an ending point \( P_2 = (3, -4) \), the direction vector \( \mathbf{d} \) is calculated as:

• \( \mathbf{d} = (3 - (-2), -4 - (-1)) = (5, -3) \)

The direction vector \( \mathbf{d} \), here \( (5, -3) \), not only indicates the path from \( P_1 \) to \( P_2 \) but also encompasses the length and orientation of that path.

Parametric Description

Understanding the parametric description of a line segment involves expressing the position using equations dependent on a parameter, commonly \( t \). By letting \( t \) vary typically between 0 and 1 for a line segment, we can describe every point along the segment.
The parametric description consolidates the initial point, direction vector, and parameter into a unified expression.

• The general formula is: \( \mathbf{r}(t) = P_1 + t \mathbf{d} \)

Translating this into separate equations for the coordinate components, we have:

• \( x(t) = -2 + 5t \)
• \( y(t) = -1 - 3t \)

These equations collectively illustrate how the points on the line segment move from the start to the end as \( t \) moves from 0 to 1.

Line Segment

A line segment is simply a part of a line that's bounded by two endpoints. It's crucial to understand when describing a line segment using parametric equations, that this method encapsulates the transition from one endpoint to the other through the parameter \( t \).
In considering the line segment from \( (-2, -1) \) to \( (3, -4) \), \( t \) is set to vary from 0 to 1, ensuring that every point on the line segment is covered as follows:

• For \( t = 0 \), we are at the starting point \( (-2, -1) \).
• For \( t = 1 \), we reach the ending point \( (3, -4) \).

Thus, the parametric equations for the segment are specifically tailored to trace the finite path between these two points. This concise approach avoids extending the line infinitely like a traditional line equation might.