Problem 32 - Algebra and Trigonometry

Question

\(29-34=\) Find parametric equations for the line with the given properties. Passing through \((12,7)\) and the origin

Step-by-Step Solution

Answer

Parametric equations: \(x = 12t\), \(y = 7t\), where \(t\) is the parameter.

Step 1: Understanding the Problem

We need to find parametric equations for a line that passes through two points: 1. The point \(12, 7\). 2. The origin, \(0, 0\). Parametric equations are a way to express the coordinates of the points on the line using a parameter. We will use the parameter \(t\).

Step 2: Identifying the Direction Vector

A direction vector for the line is found by subtracting the coordinates of the origin from the coordinates of the given point. This gives the vector \((12 - 0, 7 - 0) = (12, 7)\). So, the direction vector of the line is \((12, 7)\).

Step 3: Setting Up Parametric Equations

The parametric equations of a line can be expressed as \[ x = x_0 + at \, \ y = y_0 + bt. \] Using the point \(0, 0\) (the origin) as a point on the line \((x_0, y_0) = (0, 0)\), and the direction vector \(a, b) = (12, 7)\), the parametric equations become: \[ x = 0 + 12t = 12t \, \ y = 0 + 7t = 7t. \]

Step 4: Writing the Final Parametric Equations

The parametric equations for the line that passes through (12, 7) and the origin are: \[ x = 12t \, \ y = 7t, \] where \(t\) is the parameter.

Key Concepts

Direction Vector

A direction vector is essential when describing the path of a line. It gives the line its orientation or direction in space.

• The direction vector for a line that passes through two points can be calculated by taking the difference between the points' coordinates.
• In the example given, we found the direction vector by subtracting the origin's coordinates (0, 0) from the coordinates of point (12, 7).

This results in the direction vector \((12, 7)\).
The direction vector essentially tells us where, and in what proportion, the line moves through each coordinate axis. In this exercise, for each step along the parameter, the x-coordinate increases by 12 units, and the y-coordinate increases by 7 units.
This consistent ratio ensures that the line maintains the same direction.

Line Equations

Line equations are precise mathematical expressions to represent all points along a line. There are various forms of representing line equations, such as standard form, slope-intercept form, and parametric form. Here, we focus on the parametric form, but understanding the general concept is crucial:

• Generally, a line equation connects the line's unique direction with a fixed point that it passes through.
• This exercise displays an early introduction to the foundational aspect of line equations by exploring their parametric form, which offers a clear view into how a line extends infinitely in a defined direction.

Knowing about line equations helps us understand how lines behave under transformations and intersections.

Parametric Form

The parametric form of a line equation provides a spontaneous way to represent lines using an independent parameter, often denoted by \(t\). This flexible representation is particularly useful in many areas such as physics, computer graphics, and engineering.

• In the parametric form, each coordinate of the point on the line is expressed as a function of \(t\).
• This form is not only advantageous for its simplicity but also for making it easy to describe complex trajectories and paths.
• For instance, in our exercise, the parametric equations \(x = 12t\) and \(y = 7t\) describe all points on the line.

Each value of \(t\) corresponds to a unique point through which the line passes.
By adjusting \(t\), we can traverse the entire line, moving seamlessly through all its points.