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Chapter 11: Problem 24

Find parametric equations for the line with the given properties. Slope \(-2,\) passing through \((-10,-20)\)

Short Answer

Expert verified
x(t) = -10 + t and y(t) = -20 - 2t are the parametric equations of the line.

Step by step solution

01

Understand the line equation

The general parametric equations for lines involve a direction vector and a point through which the line passes. In 2-dimensional space, this involves finding expressions for both the x and y coordinates in terms of a parameter, usually denoted as \( t \).
02

Determine direction vector

The slope given is \(-2\). The direction for the line can be represented by the vector \((1, -2)\). This means for every unit increase in \(x\), \(y\) decreases by 2 because of the slope \(-2\).
03

Use point to form equations

We have the point \((-10, -20)\). Using this, we can form the equations as follows: **Equation for x:** Start with \(-10\) and add the change in \(x\), which is \(1 \times t\), so \(x = -10 + t\). **Equation for y:** Start with \(-20\) and add the change in \(y\), which follows the slope \(-2\), so \(y = -20 - 2t\).
04

Write final parametric equations

Using the point-slope concept from the previous steps, the parametric equations of the line are:\[\begin{align*}x(t) & = -10 + t \y(t) & = -20 - 2t\end{align*}\]

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Key Concepts

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

Direction Vector
In the context of parametric equations, the direction vector is a key component that defines how a line progresses through a coordinate plane. Imagine a line as a path that starts at a given point and follows a specific direction indefinitely. This trajectory is captured by the direction vector.

A direction vector shows the relative changes in the x and y directions for each step along the line. In our example, the slope of y given is y y a 2, so we choose the vector The Direction Vector: (1, -2) An important thing to note:
  • The direction vector (1, -2) tells us that as x increases by 1 unit, y decreases by 2 units.
  • This vector is crucial because it maintains the given slope across the line.
Slope
The slope refers to the steepness or incline of a line on a coordinate plane. It is mathematically defined as the 'rise over run', which in simpler terms is the change in y over the change in x.

For any two points on a line, the slope can be calculated as:\[\text{Slope} (m) = \frac{\Delta y}{\Delta x}\]where \( \Delta y \) is the change in y-values and \( \Delta x \) is the change in x-values.
The Negative Slope: -2
  • Our line has a slope of 2. This indicates for every 1 unit the line moves horizontally, it moves 2 units down vertically.
  • A negative slope means the line descends as it moves to the right, visualizing the idea of a hill going downhill.
Point-Slope Form
The concept of point-slope form is a powerful tool for writing equations of lines. It uses a known point and the slope to define the line's equation efficiently.
The formula for point-slope form is:\[(y - y_1) = m(x - x_1)\]where:
  • \(m\) is the slope
  • \((x_1, y_1)\) are the coordinates of the known point.
This form allows us to quickly derive the equation of a line without needing two separate points.
Applying Point-Slope FormIn this example:
  • We have a slope: 2
  • We have a point: (-10, -20)
  • Using the point-slope form helps us transition directly into creating parametric equations.
2-Dimensional Space
2-dimensional space (or 2D space) refers to a plane consisting of two axes: x and y. These axes form the basis for plotting points and visualizing geometric shapes, like lines, circles, and parabolas.
Understanding 2D Space
  • The x-axis runs horizontally, while the y-axis runs vertically.
  • Any point in this space is denoted as \((x, y)\), showing its position relative to both axes.
  • In this problem, our line with slope 2 suggests a downward trend, with T representing progress along x and y.
Working within this framework of x and y allows us to use parametric equations for tracing paths, like lines, giving us a deeper understanding of spatial relationships.

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Most popular questions from this chapter

Find an equation for the ellipse that satisfies the given conditions. Foci \((0, \pm 2),\) length of minor axis 6

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=2 t+1, \quad y=\left(t+\frac{1}{2}\right)^{2} $$

If a projectile is fired with an initial speed of \(v_{0} \mathrm{ft} / \mathrm{s}\) at an angle \(\alpha\) above the horizontal, then its position after \(t\) seconds is given by the parametric equations $$ x=\left(v_{0} \cos \alpha\right) t \quad y=\left(v_{0} \sin \alpha\right) t-16 t^{2} $$ (where \(x\) and \(y\) are measured in feet). Show that the path of the projectile is a parabola by eliminating the parameter \(t\)

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\cos ^{3} t, \quad y=\sin ^{3} t, \quad 0 \leq t \leq 2 \pi $$

\begin{array}{l}{1-22 \text { a pair of parametric equations is given. }} \\\ {\text { (a) Sketch the curve represented by the parametric equations. }} \\\ {\text { (b) Find a rectangular-coordinate equation for the curve by }} \\\ {\text { eliminating the parameter. }}\end{array} $$ x=\sqrt{t}, \quad y=1-t $$
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