Problem 26 - Precalculus: Mathematical 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 the Points

To write the parametric equations of a line, we first identify the two given points which are \( (12, 7) \) and the origin \( (0, 0) \).

Step 2: Determine the Direction Vector

Calculate the direction vector, which can be found by subtracting the coordinates of the origin from the point \( (12, 7) \). This gives us the vector \[ \vec{d} = (12 - 0, 7 - 0) = (12, 7) \].

Step 3: Formulate Parametric Equations

Using the point \( (0, 0) \) and the direction vector \( \vec{d} = (12, 7) \), create the parametric equations for the line. The equations are \[ x = 0 + 12t, \quad y = 0 + 7t \]where \( t \) is the parameter.

Key Concepts

Direction Vector

In mathematics, a **direction vector** is essential when describing a line in space. Imagine a line extending infinitely in both directions. The direction vector indicates the direction in which this line extends. It's like drawing an arrow along the line and seeing where it points. In the given exercise, to find a direction vector for a line passing through points

• (12, 7)
• (0, 0)

we subtract the coordinates of the starting point from the endpoint. This is simply taking the endpoint's components and subtracting each by the respective component of the starting point. Here, the direction vector \[\vec{d} = (12 - 0, 7 - 0) = (12, 7)\] is calculated. This direction vector

• indicates how far and in which direction you would travel from the origin to reach the point (12, 7)
• conveys both the line's slope and its orientation in the plane

Parametric Form

The **parametric form** of a line gives you an advanced way to describe the line's path by treating each coordinate as a function of a single variable called the parameter, often denoted by \( t \).
For a line that runs through a specific point, the parametric equation articulates the progression from this fixed point along the direction given by the direction vector. In the solved problem, the line passes through the starting point, which is the origin \((0, 0)\).
The parametric equations are given as:

• \( x = 0 + 12t \)
• \( y = 0 + 7t \)

Here,

• \( t \) represents how far along the line you go
• each equation accounts for advance along \(x\) and \(y\) axes respectively

This form is quite useful, especially in computational applications, because it allows movement along the line to be represented in a straightforward way.

Coordinate Subtraction

**Coordinate subtraction** lays the groundwork for finding direction vectors and is a fundamental operation in vector mathematics that helps determine the difference between two points. This process involves taking each coordinate of the initial point and subtracting the corresponding coordinate of another point.In the example, to get from the origin \((0, 0)\) to the point \((12, 7)\), you perform the following operations:

• Subtract the x-coordinate of start from the x-coordinate of the destination:\[12 - 0 = 12\]
• Subtract the y-coordinate of start from the y-coordinate of the destination:\[7 - 0 = 7\]

Understanding this operation:

• helps visualize the shift or movement required in each dimension
• forms the basis for calculations in more intricate math problems

Coordinate subtraction not only assists in gaining the direction in which a line runs but is also widely applicable in fields of geometry and physics for tasks involving translations and transformations.