Problem 25 - Algebra and Trigonometry

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 direction vector of the line, subtract the coordinates of the two given points. For points \((6,7)\) and \((7,8)\), the direction vector \(\mathbf{d}\) is given by \((7-6, 8-7) = (1, 1)\).

Step 2: Set Up the Parametric Equations

Using the direction vector \((1, 1)\) and the point \((6,7)\), we set up the parametric equations by adding the direction vector times a parameter \(t\) to the point. The parametric equations are: \[ x = 6 + 1t \] \[ y = 7 + 1t \] These express \(x\) and \(y\) in terms of the parameter \(t\).

Step 3: Verify the Parametric Equations

Substitute \(t = 0\) into the equations to verify that they pass through the point \((6,7)\). We get \(x = 6\) and \(y = 7\), so it passes through \((6,7)\). For point \((7,8)\), substitute \(t = 1\): \[ x = 6 + 1 \cdot 1 = 7 \] \[ y = 7 + 1 \cdot 1 = 8 \] This shows the equations also pass through \((7,8)\).

Key Concepts

Direction Vector

When dealing with parametric equations of a line, one of the first steps is to determine the direction vector. The direction vector indicates the direction in which the line is pointing. To find it, you need to subtract the coordinates of one point from the other. For instance, if you have two points,

• (6,7) and (7,8),

the direction vector, \( \mathbf{d} \), is given by the difference of these coordinates:

• \((7-6, 8-7) = (1, 1)\).

This vector \((1, 1)\) essentially tells us how to move from one point to another along the line. It’s important because it essentially defines the slope, or direction, of the line in a parametric equation.
This direction vector will later help in generating parametric forms of equations that define the full line.

Parametric Form

The parametric form of a line is a powerful way to express the equation of a line. It uses the direction vector and a point on the line to create a system of equations. With the direction vector \((1, 1)\) and the point \((6,7)\), you can create the parametric equations as follows:

• \( x = 6 + 1t \)
• \( y = 7 + 1t \)

Here, \( t \) is a parameter, which can be any real number. These equations convey that the line can be described by moving along the direction vector from the initial point \((6, 7)\).
As \( t \) changes, the point \((x, y)\) moves along the line. This parametric form also makes it easy to find specific points on the line by choosing specific values of \( t \).

Verify Equations

Verifying parametric equations is an essential step to ensure accuracy. This involves checking if given points actually satisfy the equations. Start by substituting a simple value of the parameter, such as \( t = 0 \), into the equations.

• For \( t = 0 \):
  \[ x = 6 + 1 \cdot 0 = 6 \]
  \[ y = 7 + 1 \cdot 0 = 7 \]
  This confirms it passes through (6,7).

Then substitute another value, say \( t = 1 \), to test the other point:

• For \( t = 1 \):
  \[ x = 6 + 1 \cdot 1 = 7 \]
  \[ y = 7 + 1 \cdot 1 = 8 \]
  This shows the line also goes through (7,8).

By substituting different \( t \) values, you ensure the validity of the equations and confirm that they accurately describe the line passing through the given points. It's a simple yet vital step to verify that the math checks out and the parametric equations are correctly formed.