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Chapter 9: Problem 33

Find parametric equations describing the given curve. The line segment from (-2,4) to (6,1)

Short Answer

Expert verified
The parametric equations describing the line segment from (-2,4) to (6,1) are \( x = -2 + 8t \) and \( y = 4 - 3t \), where \( 0 \leq t \leq 1 \).

Step by step solution

01

Set the Initial Points

Let's start by setting our initial point, which will be (-2,4). We will derive the parametric equation from this point.
02

Define the Distance Covered

Next, we need to define the distance covered to move from the start point to the end point by subtracting the initial points from the final points. This gives (6 - (-2), 1 - 4) = (8, -3).
03

Define the Parameter

Let \( t \) be a parameter satisfying \( 0 \leq t \leq 1 \). When \( t = 0 \), we are at the initial point, and when \( t = 1 \), we've reached the final point.
04

Express the Variables in Terms of the Parameter

We can express the lines in terms of the parameter \( t \) as follows: \( x = -2 + 8t \) and \( y = 4 - 3t \). These are the parametric equations we were asked to provide.

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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 a part of a line bounded by two distinct endpoints. Unlike an infinite line, a line segment has a defined start and end. In our exercise, the segment connects the point (-2, 4) to the point (6, 1).
  • **Endpoints:** The defined points at the start and end of a line segment.
  • **Length:** The distance between these endpoints. You can use the distance formula to calculate this in a two-dimensional space.
Line segments are crucial in mathematics as they can represent various real-world situations, such as roads or connecting lines in diagrams. In this exercise, we focus on computing parametric equations to describe this line mathematically.
Parameterization
Parameterization involves expressing a set of equations or functions based on a parameter. In geometry, when describing curves or line segments, we convert problems from conventional equations to those involving a single variable parameter, often symbolized as \( t \).
In the example given:
  • The parameter \( t \) simplifies transitioning from the starting point to the endpoint.
  • The value \( t = 0 \) represents the initial point (-2, 4).
  • The value \( t = 1 \) represents the final point (6, 1).
The range of \( t \), specifically \( 0 \leq t \leq 1 \), ensures the parameter accurately traces the segment from start to end without extending beyond either point. Parameterization creates a streamlined approach for navigation and integration of line segments in broader mathematical problems.
Point Coordinates
Coordinates are numerical values that specify the exact location of a point in a space. In a two-dimensional plane, we use pairs of numbers, typically represented as \( (x, y) \), to fixate the position.
When defining the line segment from the previous sections, knowing the coordinates of our endpoints is essential:
  • The first point is \((-2, 4)\).
  • The second point is \((6, 1)\).
These coordinates become the basis for our parametric equations, assisting in understanding how the line segment is traced as \( t \) changes. By manipulating these coordinates using algebra, one can translate the segment into its parametric form, providing a dynamic method to navigate point locations.

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Most popular questions from this chapter

In exercises find the area enclosed by the given curve. $$\left\\{\begin{array}{l} x=\frac{1}{2} \cos t-\frac{1}{4} \cos 2 t \\ y=\frac{1}{2} \sin t-\frac{1}{4} \sin 2 t \end{array}\right.$$

Find parametric equations of the conic sections. $$\frac{(x+1)^{2}}{9}+\frac{(y-1)^{2}}{4}=1$$

Graph the conic section and find an equation. All points such that the sum of the distances to the points (3,1) and (-1,1) equals 6

In exercises parametric equations for the position of an object are given. Find the object's velocity and speed at the given times and describe its motion $$\left\\{\begin{array}{l} x=2 \cos 2 t \\ y=2 \sin 2 t \end{array}\right.$$

Find a polar equation corresponding to the given rectangular equation. $$y=3$$
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