Problem 81 - Thomas Calculus

Question

In Exercises \(81-86,\) find a parametrization for the curve. the line segment with endpoints \((-1,-3)\) and \((4,1)\)

Step-by-Step Solution

Answer

Parametrization: \((x, y) = (-1 + 5t, -3 + 4t)\), \(t \in [0, 1]\).

Step 1: Determine the Direction Vector

To parametrize the line segment, first find the direction vector by subtracting the coordinates of the starting point from the endpoint. The direction vector \( \mathbf{d} \) is given by \((4 - (-1), 1 - (-3)) = (5, 4)\).

Step 2: Establish Parametric Equations

Use the starting point \((-1, -3)\) and the direction vector \((5, 4)\) to write the parametric equations. Let \(x = -1 + 5t\) and \(y = -3 + 4t\).

Step 3: Restrict the Parameter

Determine the appropriate range for \(t\). As \(t = 0\) gives the starting point \((-1, -3)\) and \(t = 1\) gives the endpoint \((4, 1)\), limit \(t\) to \([0, 1]\) for the line segment.

Key Concepts

Parametrization of curves

In calculus, the concept of parametrization allows us to express a curve using a set of equations. This is particularly useful for generating line segments, circles, and various other geometries. By using parameters, typically denoted as a variable like \( t \), we can express different parts of a curve in a more manageable form. This approach is beneficial when dealing with complex curves or when performing integration.

• Parametrization breaks down a curve into simpler, coordinate-based representations.
• It provides a method to describe motion along a path.
• By adjusting the parameter, we can move along the curve from one endpoint to another.

An example of this is finding a way to express the line segment between two points \((-1, -3)\) and \((4, 1)\). Doing so involves determining a vector direction and creating an equation that encompasses all coordinates between these points.
Ultimately, parametrization is not only helpful in math but also in disciplines like physics and engineering, where understanding movement and position is crucial.

Line segments

A line segment is the portion of a line that connects two endpoints. Unlike infinite lines, line segments have a defined beginning and end. This makes them a fundamental component in geometry as well as in calculus when it comes to parametrizing curves.

• Every line segment can be expressed using endpoints that define its boundaries.
• Line segments do not extend beyond their endpoints, maintaining a fixed length and finite path.

In this exercise, we identified a line segment stretching from \((-1, -3)\) to \((4, 1)\). Finding its parametrization involves determining how to smoothly transition from the starting point to the endpoint by using a parameter, typically denoted as \( t \).
Understanding line segments helps in dividing larger shapes into simpler, more manageable parts for analysis or calculation.

Direction vectors

Direction vectors play a crucial role in parametrizing line segments by indicating the direction and magnitude of the segment. The vector essentially guides us from one point to another by providing the necessary "+step size" for transformation.

• A direction vector is derived by subtracting the coordinates of the start point from those of the endpoint.
• It represents the journey in terms of horizontal and vertical movement needed to reach the endpoint from the start.
• In our example: direction vector \( (5, 4) \) indicates moving 5 units right and 4 units up from \((-1, -3)\) towards \((4, 1)\).

Using direction vectors ensures that our parametric equations accurately represent the line segment by adjusting the rate of movement across the segment. Given as part of a vector equation, it enables precise control over the movement direction and the extent that each parameter \( t \) covers within its specified range.