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Chapter 11: Problem 61

Show that parametric equations for a line passing through the points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) are $$ \begin{array}{l} x(t)=\left(x_{2}-x_{1}\right) t+x_{1} \\ y(t)=\left(y_{2}-y_{1}\right) t+y_{1} \quad-\infty

Short Answer

Expert verified
The line moves from \(\left(x_{1}, y_{1}\right)\) to \(\left(x_{2}, y_{2}\right)\) as t increases.

Step by step solution

01

- Understand the line equation

The parametric equations describe a line where each point on the line is represented by a parameter, typically denoted as t.
02

- Define the parametric equations

Given two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\), the parametric equations are: \[x(t)=\left(x_{2}-x_{1}\right)t+x_{1} \] \[y(t)=\left(y_{2}-y_{1}\right)t+y_{1}\]
03

- Find the changes in variables

Let \(\Delta x = x_{2} - x_{1}\) and \(\Delta y = y_{2} - y_{1}\). These represent the differences in the x and y coordinates between the two points.
04

- Substitute into the equations

Substitute \(\Delta x\) and \(\Delta y\) into the parametric equations: \[x(t) = \Delta x \cdot t + x_{1}\] \[y(t) = \Delta y \cdot t + y_{1}\]
05

- Verify the points

When \(t = 0\), the parametric equations give \[x(0) = x_{1}\] and \[y(0) = y_{1}\]. When \(t = 1\), they give \[x(1) = x_{2}\] and \[y(1) = y_{2}\]. This shows the equations pass through both points.
06

- Determine the orientation

The orientation of the line is given by the direction from \(\left(x_{1}, y_{1}\right)\) to \(\left(x_{2}, y_{2}\right)\). As t increases, the line moves from \(\left(x_{1}, y_{1}\right)\) towards \(\left(x_{2}, y_{2}\right)\).

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Key Concepts

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

Understanding Parametric Equations
Parametric equations are a powerful tool in coordinate geometry. They describe a line using parameters, typically denoted as t, which varies over real numbers.
These equations provide a way to describe the coordinates of any point on the line. In this case, we use t to represent different points on the line. The general form for a line passing through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by:
\[ x(t) = (x_2 - x_1)t + x_1 \]
\[ y(t) = (y_2 - y_1)t + y_1 \]
As t changes, it generates different points on the line, fully capturing its essence.
Coordinate Geometry and Its Application
Coordinate geometry, or analytic geometry, uses algebraic equations to represent geometric figures. A line is one of the most fundamental shapes studied in this field.
By understanding the relationship between algebraic expressions and geometric figures, you can easily handle complex problems like finding the intersection of lines, distances between points, and line segments orientation.
The parametric form of equations is particularly helpful as it sets up a bridge between algebraic equations and geometric understanding.
Line Orientation Explained
Orientation of a line is essentially the direction in which the line progresses as the parameter t increases. For the line passing through points \( (x_1, y_1) \) and \( (x_2, y_2) \), the orientation is determined by \[ \Delta x = x_2 - x_1 \]
and \[ \Delta y = y_2 - y_1 \].
The direction of movement starts from point \( (x_1, y_1) \) when t=0 to point \( (x_2, y_2) \) when t=1. As t changes from negative to positive, you move from \( (x_1, y_1) \) towards \( (x_2, y_2) \). This helps to visualize the line鈥檚 progression and orientation over its entire span.
The Role of the t Parameter
The parameter t is instrumental in the parametric equations for a line. It serves as an independent variable controlling the x and y coordinates.
Specifically, t represents a point moving along the line. For example:
  • When t = 0, you get the coordinates \[ x(0) = x_1 \] and \[ y(0) = y_1 \], the starting point.
  • When t = 1, the coordinates are \[ x(1) = x_2 \] and \[ y(1) = y_2 \], the ending point.
By varying t, you really map out the entire line between \( (-\infty, \infty) \). This makes t vital in representing the full span of the line.

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Most popular questions from this chapter

In Problems \(45-48\), use a graphing utility to graph the plane curve defined by the given parametric equations. \(x(t)=t \sin t, \quad y(t)=t \cos t, \quad t>0\)

Billy hit a baseball with an initial speed of 125 feet per second at an angle of \(40^{\circ}\) to the horizontal. The ball was hit at a height of 3 feet above the ground. (a) Find parametric equations that model the position of the ball as a function of time. (b) How long was the ball in the air? (c) Determine the horizontal distance that the ball traveled. (d) When was the ball at its maximum height? Determine the maximum height of the ball. (e) Using a graphing utility, simultaneously graph the equations found in part (a).

Use the fact that the orbit of a planet about the Sun is an ellipse, with the Sun at one focus. The aphelion of a planet is its greatest distance from the Sun, and the perihelion is its shortest distance. The mean distance of a planet from the Sun is the length of the semimajor axis of the elliptical orbit. The mean distance of Earth from the Sun is 93 million miles. If the aphelion of Earth is 94.5 million miles, what is the perihelion? Find an equation for the orbit of Earth around the Sun.

Jodi's bus leaves at 5: 30 pM and accelerates at the rate of 3 meters per second per second. Jodi, who can run 5 meters per second, arrives at the bus station 2 seconds after the bus has left and runs for the bus. (a) Find parametric equations that model the motions of the bus and Jodi as a function of time. [Hint: The position \(s\) at time \(t\) of an object having acceleration \(a\) is \(\left.s=\frac{1}{2} a t^{2} .\right]\) (b) Determine algebraically whether Jodi will catch the bus. If so, when? (c) Simulate the motion of the bus and Jodi by graphing simultaneously the equations found in part (a).

Find the work done by a force of 80 pounds acting in the direction of \(50^{\circ}\) to the horizontal in moving an object 12 feet from (0,0) to (12,0) . Round to one decimal place.
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