Problem 25 - Precalculus: Mathematical 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: Determine the direction vector

The direction vector of the line can be found by subtracting the coordinates of the given points. If we call the points \((x_1, y_1) = (6,7)\) and \((x_2, y_2) = (7,8)\), the direction vector \(\mathbf{v}\) is given by \((x_2 - x_1, y_2 - y_1) = (7 - 6, 8 - 7) = (1, 1)\). Thus, the direction vector is \( (1, 1) \).

Step 2: Set up the parametric equations

Using the point \((6,7)\) as a point on the line and the direction vector \( (1, 1) \), we can write the parametric equations for the line. The formula for a parametric equation of a line is \((x, y) = (x_0, y_0) + t \cdot (a, b)\), where \((x_0, y_0)\) is a point on the line and \((a, b)\) is the direction vector. Substituting in our known values gives us \( x = 6 + t \cdot 1 = 6 + t \) and \( y = 7 + t \cdot 1 = 7 + t \).

Step 3: Write the final parametric equations

The parametric equations for the line passing through \((6,7)\) and \((7,8)\) are: \( x = 6 + t \) and \( y = 7 + t \), where \( t \) is a real number parameter.

Key Concepts

Direction Vector

When working with parametric equations to find the equation of a line, one key concept is the direction vector. This vector tells us the "direction" in which the line moves through space. If you imagine arrows pointing straight through the points on the line, the direction vector is essentially one of those arrows.

To find the direction vector between two points, you subtract the coordinates of these points. Say you have two points, \(A(x_1, y_1)\) and \(B(x_2, y_2)\). The direction vector, denoted \(\mathbf{v}\), is calculated as \( (x_2 - x_1, y_2 - y_1)\).

Here is why the direction vector is important:

• It tells us how to "travel" from one point on the line to another, maintaining the same direction.
• It's the backbone of creating parametric equations, giving us a way to extend the line infinitely in both directions.

By understanding and finding the direction vector, you can easily move on to write the parametric equations for a line.

Line Equations

Line equations are fundamental in coordinate geometry, serving as mathematical descriptions of straight lines. There are different forms of line equations, and one useful form based on direction vectors is the parametric form.

In parametric equations, a line is described using a parameter, typically \(t\), which represents a movement along the line. This form is flexible and descriptive. It is structured as \( (x, y) = (x_0, y_0) + t \cdot (a, b)\), where:

• \((x_0, y_0)\) is a known point on the line.
• \((a, b)\) is the direction vector.

For example, if you know a point \((6,7)\) and the direction vector is \((1, 1)\), the line's parametric equations become \(x = 6 + t\) and \(y = 7 + t\). Here you can imagine \(t\) representing how far you "travel" along the line from the point \((6,7)\).

This approach gives a straightforward way to handle lines, including when they extend infinitely in both directions.

Coordinate Geometry

Coordinate geometry, sometimes called analytic geometry, merges algebra with geometry by using algebraic equations to describe geometric figures. This rich field allows us to understand and analyze shapes and sizes of different geometrical figures mathematically.

Key concepts in coordinate geometry include and enable the understanding of:

• Points: Defined by coordinates, such as \((x, y)\), representing positions in a plane.
• Lines: Can be described by equations, including parametric equations, which are very helpful for understanding the infinite extension of lines in graphical space.
• Development of systematic methods to derive one concept from another, such as the transition from points and vectors to full line equations.

These principles form the backbone of coordinate geometry, providing tools to study distance, slope, and point location based on algebraic representations, making it easier to visualize and solve complex geometrical problems.