Problem 31 - Precalculus Mathematics for Calculus

Question

Find parametric equations for the line with the given properties. Passing through \((6,7)\) and \((7,8)\)

Step-by-Step Solution

Answer

The parametric equations are \(x = 6 + t\) and \(y = 7 + t\).

Step 1: Identify the Direction Vector

To find the parametric equations of the line, we need to determine the direction vector. The line passes through the points \((6,7)\) and \((7,8)\). The direction vector \(\mathbf{d}\) can be found by subtracting the coordinates of these points: \(\mathbf{d} = (7 - 6, 8 - 7) = (1, 1)\).

Step 2: Determine a Point on the Line

Any point on the line can act as a reference point. We are given two points, \((6,7)\) and \((7,8)\). We can choose either as a point on the line. Let's choose \((6,7)\).

Step 3: Write the Parametric Equations

Using the point \((6,7)\) and the direction vector \((1,1)\), we write the parametric equations for the line. If \(t\) is a parameter, then the parametric equations are: \[ x = 6 + 1t \]\[ y = 7 + 1t \] This describes the line as it passes through the chosen point with the given direction vector.

Key Concepts

Direction Vector

In parametric equations, a direction vector provides crucial information to define the path of a line. It essentially indicates how the line runs through the coordinate plane.
The direction vector is denoted as \(\mathbf{d}\). To find it, identify two points on the line. For instance, consider points \((6,7)\) and \((7,8)\). By subtracting their coordinates, you obtain \((7 - 6, 8 - 7)\), which simplifies to \((1, 1)\).

• **Direction vector**: Indicates the direction along a line.
• **Calculate**: Subtract coordinates of two given points.

This vector \((1, 1)\) shows us that as you move along the line, both x and y increase by 1 unit for each step in the parameter \(t\). Such vectors are essential in describing the behavior of a line.

Points on a Line

In mathematics, identifying points on a line is fundamental to constructing parametric equations. We typically use one or more known points to help describe the line's trajectory.
For example, consider the points \((6,7)\) and \((7,8)\) mentioned in the original problem. Either of these can serve as a reference point when writing the parametric form of a line.

• **Chosen Point**: Acts as a fixed point on the line for simplicity in equations.
• **Arbitrary Choice**: Either point can be selected without affecting the accuracy.

In our example, using \((6,7)\), we ground our line in space, allowing for the dynamic element, \(t\), to dictate the reach along our direction vector.

Parametric Form

The parametric form of a line lets us write equations that describe how each coordinate of a point changes with respect to a parameter \(t\). It captures the dynamic essence of a line in motion as \(t\) varies.
Utilizing our earlier identified direction vector \((1,1)\) and chosen point \((6,7)\), we can express the parametric equations as:
\[x = 6 + 1t\]
\[y = 7 + 1t\]

• **Parametric Equation**: Expresses the x and y coordinates as functions of \(t\).
• **Line Representation**: Specifies coordinates changes in tandem with \(t\).

Such representation is highly beneficial in mathematics due to its flexibility and clarity in describing line paths.