Problem 22 - Thomas Calculus

Question

Find a parametrization for the curve. the line segment with endpoints \((-1,3)\) and \((3,-2)\)

Step-by-Step Solution

Answer

The parametrization is \\(x(t) = -1 + 4t, \, y(t) = 3 - 5t\\) for \\(t \\in [0, 1]\\).

Step 1: Identify the Line Segment

We begin by identifying the endpoints of the line segment, which are \((-1, 3)\) and \(3, -2)\). We recognize that a linear path between two points can be represented parametrically using these endpoints.

Step 2: Determine the Vector Representation

Calculate the direction vector by subtracting the first endpoint from the second: \(\langle 3 - (-1), -2 - 3 \rangle = \langle 4, -5 \rangle\). This vector will be used to describe the difference along the line segment.

Step 3: Parametrize the Line Segment

Use the parametrization formula \(\mathbf{r}(t) = \mathbf{a} + t\mathbf{d}\), where \(\mathbf{a}\) is the starting point and \(\mathbf{d}\) is the direction vector. For this segment, it becomes \(\mathbf{r}(t) = (-1, 3) + t\langle 4, -5 \rangle\) for \(t \in [0, 1]\).

Step 4: Write the Parametric Equations

Convert the vector equation into parametric equations for each coordinate: \(x(t) = -1 + 4t\) and \(y(t) = 3 - 5t\), where \(t \in [0, 1]\). These define the line's position at any point \(t\) within the interval.

Step 5: Verify the Endpoints

Verify that \(t = 0\) yields the start point \((-1, 3)\) and \(t = 1\) yields the endpoint \((3, -2)\). Substituting \(t = 0\) gives: \(x(0) = -1\) and \(y(0) = 3\). Substituting \(t = 1\) gives: \(x(1) = 3\) and \(y(1) = -2\). This confirms the correct parametrization.

Key Concepts

Line Segment Parametrization

One fundamental way to describe a line segment mathematically is through parametrization. Imagine a line on a graph. A line segment is simply a portion of this infinite line, specifically between two endpoints.
To represent this, we use a parameter, typically denoted as \( t \), which varies over a specified range to traverse the segment. In the case of a straight line segment between points \( (-1, 3) \) and \( (3, -2) \), we use \( t \) in the range \([0, 1]\).

• The initial endpoint corresponds to \( t=0 \).
• The final endpoint corresponds to \( t=1 \).

With this approach, as \( t \) transitions from 0 to 1, the line segment is "drawn" through a linear path, making every intermediate point between the endpoints easy to calculate.

Parametric Equations

To fully capture the progression along a line, we break down the vector equation into simpler, separate functions for each coordinate axis—these are known as parametric equations. They allow us to describe exactly where any point on the segment is, given a specific \( t \) value.

For our example, the vector equation \( \mathbf{r}(t) = (-1, 3) + t \langle 4, -5 \rangle \) can be split into two functions:

• \( x(t) = -1 + 4t \): This equation determines the \( x \)-coordinate.
• \( y(t) = 3 - 5t \): This equation determines the \( y \)-coordinate.

Both these equations combined tell us the exact position on the plane at any point. By plugging values of \( t \) from 0 to 1 into each equation, we determine the position across the line. Parametric equations are especially helpful in animations and simulations, where movements need to be calculated precisely.

Direction Vectors

An essential component in parametrizing a line segment is the direction vector. It's like the driving force that points and moves the segment from one endpoint to another. To find it, we subtract the coordinates of the initial point from the final point:

\[ \text{Direction Vector} = \langle x_2-x_1, y_2-y_1 \rangle\]For the described line segment, this calculation goes as:

• In the \( x \)-direction: \( 3 - (-1) = 4 \)
• In the \( y \)-direction: \( -2 - 3 = -5 \)

This gives us the direction vector \( \langle 4, -5 \rangle \). It's responsible for defining the path's slope and ensures a consistent travel along the segment. Basically, it connects the change in \( x \) and \( y \) needed to get from start to finish, demonstrating how much each parameter impacts the segment's path.