Problem 32 - Precalculus Mathematics for Calculus

Question

Find parametric equations for the line with the given properties. Passing through \((12,7)\) and the origin

Step-by-Step Solution

Answer

The parametric equations are \(x = 12t\) and \(y = 7t\).

Step 1: Identify Points

Start by identifying the two given points. The points are the origin \((0, 0)\) and \((12, 7)\).

Step 2: Calculate Direction Vector

Find the direction vector of the line by subtracting the coordinates of the origin from the coordinates of the point \((12, 7)\). This results in the vector \((12 - 0, 7 - 0) = (12, 7)\).

Step 3: Set Up Parametric Equations

Using the direction vector \((12, 7)\) and a point on the line, let's use the origin \((0, 0)\), we can write the parametric equations as follows: \(x = 0 + 12t\) and \(y = 0 + 7t\).

Step 4: Write Final Parametric Equations

Thus, the parametric equations of the line are \[x = 12t\] and \[y = 7t\], where \(t\) is a parameter.

Key Concepts

Direction Vector

A direction vector is essential in defining and understanding a line in parametric equations. For any line, a direction vector helps indicate the line's orientation or path. These vectors are defined by subtracting the coordinates of one point on the line from another.
For example, given two points, the origin (0, 0) and (12, 7), the direction vector is found by evaluating the differences (12 - 0, 7 - 0), resulting in the vector (12, 7).
This vector (12, 7) is crucial because it tells us how much the line moves horizontally and vertically as it progresses. In this case, for every step horizontally by 12 units, we move vertically by 7 units.

• Direction vectors are fundamental for parametric equations, which form the building blocks of coordinate geometry.
• They help us express the line mathematically, ensuring clarity and precision.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, allows us to study geometry using a coordinate system. This branch of mathematics bridges algebra with geometry, enabling us to describe geometric shapes and properties using algebraic equations.
In the context of our exercise, coordinate geometry helps us derive parametric equations for a line given certain conditions. By plotting points and direction vectors in coordinate geometry, we can visually and algebraically find properties of the line such as slope, and intercepts.
Using systems like the Cartesian plane (2D plane with x and y axes), we can translate geometrical problems into algebraic formulations, making complex problems more manageable.

• Coordinate geometry simplifies the study of distances, angles, and relationships between various geometrical figures.
• It utilizes the idea of direction vectors to provide precise descriptions of lines and curves.

Line Through Origin

A line through the origin is a unique and significant aspect of geometry. This type of line passes through the point (0, 0), which serves as a natural reference point in the coordinate system.
For any such line, the equation can be simplified as it eliminates the constant terms (or intercepts) commonly found in line equations. The parametric equation for a line through the origin only involves direction vectors and parameters.
For example, in our exercise, the line runs through (0, 0) and uses the direction vector (12, 7). Hence, the parametric equations are x = 12t and y = 7t, where t is the variable parameter.

• Lines through the origin are proportional; when written as fractions, x/y is equal to the direction vector x/y.
• This concept is crucial for simplifying problems in physics and engineering, where the origin often represents an equilibrium or a starting point for measurements.

Points in Geometry

In geometry, points serve as the most foundational elements. They have no dimension; they only indicate position. In parametric equations, specific points can help define vectors and lines.
When working with lines and their equations, identifying key points like the origin or specific targets such as (12, 7) aids in the mathematical description of the line.
Given two points, one can determine direction vectors, slopes, and ultimately, line equations. These points simplify complex geometric and algebraic interpretations, making problems more straightforward.

• Identifying critical points reduces uncertainty while solving geometric problems.
• Points, when linked, form lines; and lines when combined, create shapes, making points pivotal in all geometric constructions.
• In parametric form, points are essential for setting initial conditions and forming direction vectors, which are fundamental for describing the line's path.