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Chapter 11: Problem 32

In Exercises \(31-36,\) find a parametrization for the curve. the line segment with endpoints \((-1,3)\) and \((3,-2)\)

Short Answer

Expert verified
The parametrization is \(x(t) = -1 + 4t\), \(y(t) = 3 - 5t\), with \(0 \leq t \leq 1\).

Step by step solution

01

Understand the Problem

The goal is to find a function that describes every point on the line segment between two given endpoints, \(-1, 3\) and \(3, -2\) , using a parameter, typically denoted as \(t\).
02

Set Up the Parametric Equations

To parametrically describe a line segment, we use the formulas: \(x(t) = x_0 + t(x_1 - x_0)\) and \(y(t) = y_0 + t(y_1 - y_0)\), where \(t\) ranges from 0 to 1, and \( (x_0, y_0)\) and \( (x_1, y_1)\) are the endpoints of the line segment.
03

Substitute Endpoint Coordinates

Substitute \((x_0, y_0) = (-1, 3)\) and \( (x_1, y_1) = (3, -2)\) into the parametric equations. For the \(x\) component: \(x(t) = -1 + t(3 - (-1)) = -1 + 4t\).For the \(y\) component: \(y(t) = 3 + t(-2 - 3) = 3 - 5t\).
04

Write Down the Parametric Equations

The parametric equations for the line segment are \(x(t) = -1 + 4t\) and \(y(t) = 3 - 5t\), with the parameter \(t\) ranging from 0 to 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Line Segment
A line segment is the part of a line that is bounded by two distinct endpoints. Unlike an infinite line, a line segment does not extend beyond its set endpoints. It contains all the points that lie directly between its starting and end point. In geometric terms, a line segment is important because it is the shortest path connecting two points.
  • A line segment can be horizontal, vertical, or diagonal, depending on the positions of its endpoints.
  • In coordinate geometry, a line segment has a measurable length, which can be calculated using the distance formula.
  • When represented on a graph, the endpoints of a line segment differentiate it from a line, which extends infinitely in both directions.
Understanding a line segment helps us to express portions of more complex curves and shapes using easier mathematical descriptions.
Parametrization
Parametrization is the process of defining a mathematical concept using parameters. In the context of a line segment, it involves expressing the coordinates of every point on the segment as a function of one or more variables called parameters.
To parametrize a line segment, we find a set of equations that describe how a point moves from one endpoint to another. The typical parameter used is denoted as \(t\), which varies usually from 0 to 1.
  • When \(t = 0\), the point is at the start of the segment.
  • When \(t = 1\), the point reaches the end of the segment.
  • The values of \(t\) in between correspond to points in between the two endpoints.
Parametrization simplifies the process of defining and manipulating line segments in computational and algebraic scenarios.
Coordinates
Coordinates describe the specific conditions of a point in a geometric space, typically in terms of a set of numerical values. In the Cartesian coordinate system, any point on a plane is represented by an ordered pair \((x, y)\), where \(x\) is the horizontal position and \(y\) is the vertical position.
For line segments, these coordinates play a critical role in identifying the exact location and direction of the segment.
  • The change in these coordinates between two endpoints dictates the slope and orientation of the line segment.
  • By using coordinates, we can calculate various aspects like the midpoint, length, and section division of the segment.
  • They are foundational for setting up parametric equations to map out the entire line segment.
Understanding how to work with coordinates is vital for solving geometry problems involving line segments and other shapes.
Endpoints
Endpoints are the two distinct points that denote the boundaries of a line segment. They are the beginning and the end of the segment, and are essential for defining its overall position and length.
  • The coordinates of the endpoints allow us to determine the properties of the line segment, such as its length using the distance formula \( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \).
  • These points serve as references for creating the parametric equations that describe all points lying on the line segment.
  • In problems involving the parametrization of a line segment, defining the segment's endpoints clearly is crucial for ensuring accuracy.
Consequently, understanding endpoints helps in accurately defining line segments and further simplifies problems involving various mathematical and geometric transformations.

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