Problem 24 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: 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 \( t \).

Step 2: 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 \(-2\).

Step 3: 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 \(y = -20 - 2t\).

Step 4: Write final parametric equations

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

Key Concepts

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.