Problem 14 - Algebra and Trigonometry

Question

Find parametric equations for the line that passes through the points \(P\) and \(Q .\) $$ P(12,16,18), \quad Q(12,-6,0) $$

Step-by-Step Solution

Answer

The parametric equations are: \(x = 12\), \(y = 16 - 22t\), \(z = 18 - 18t\).

Step 1: Determine the direction vector

To find the parametric equations, we start by determining the direction vector of the line, which can be found by subtracting the coordinates of point \(P\) from point \(Q\). The direction vector \(\vec{d}\) is given by \(\vec{d} = Q - P\). Calculate it as follows: \(\vec{d} = (12 - 12, -6 - 16, 0 - 18) = (0, -22, -18)\).

Step 2: Set up the parametric equations

The parametric equations of a line are given by: \(x = x_0 + at\), \(y = y_0 + bt\), \(z = z_0 + ct\), where \((x_0, y_0, z_0)\) are the coordinates of a point on the line (either \(P\) or \(Q\)), and \((a, b, c)\) are the components of the direction vector \(\vec{d}\). Here, we will use point \(P(12, 16, 18)\) and the direction vector \(\vec{d} = (0, -22, -18)\).

Step 3: Write down the final parametric equations

Using the point \(P(12, 16, 18)\), the parametric equations of the line are: 1. \(x = 12\) 2. \(y = 16 - 22t\) 3. \(z = 18 - 18t\).

Key Concepts

Direction Vector

A direction vector is a key concept when working with lines in geometry, especially in three-dimensional space. It provides the line with its specific direction. To find the direction vector, we use the coordinates of two points on the line. In our case, the direction vector \(\vec{d}\) can be determined by subtracting the coordinates of one point from the other.
For instance, given two points \(P(12,16,18)\) and \(Q(12,-6,0)\), the direction vector can be found as follows:

• Subtract the x-coordinates: \(12 - 12 = 0\)
• Subtract the y-coordinates: \(-6 - 16 = -22\)
• Subtract the z-coordinates: \(0 - 18 = -18\)

This results in a direction vector \(\vec{d} = (0, -22, -18)\). It provides a clear indication of how the line moves through space, showing us it does not shift in the x-direction, but descends in the y and z directions.

Three-Dimensional Space

Three-dimensional space is a geometric setting where three values (coordinates) are required to determine the positions of points or objects. In this context, each point has three coordinates which are usually represented as \((x, y, z)\). These coordinates describe how far along the point lies in each of those directions.
Working in three-dimensional space means dealing with applications that consist of length, width, and height, providing a realistic representation of physical spaces. This tri-dimensional perspective is powerful for visualizing and performing geometric computations, as it allows more complex relationships to be understood and described.
In practical situations like the line equations described here, this space lets us express lines to see how they extend and travel. Each component of the line's description (using point coordinates and a direction vector) effectively maps out where the line is located and where it leads.

Line Equations

Line equations in geometry describe the path traced by a moving point or a series of points that form a straight line. When dealing with lines in three-dimensional space, one common approach is to use parametric equations. These help provide a mathematical description of the line using variables.
Consider the parametric equations \(x = x_0 + at\), \(y = y_0 + bt\), and \(z = z_0 + ct\), where:

• \((x_0, y_0, z_0)\) represent the coordinates of a point on the line.
• \((a, b, c)\) are components of the direction vector.
• \(t\) is a parameter that varies over all real numbers.

For example, using point \(P(12, 16, 18)\) and direction vector \(\vec{d} = (0, -22, -18)\), the parametric equations become:

• \(x = 12\)
• \(y = 16 - 22t\)
• \(z = 18 - 18t\)

This setup allows one to calculate any point along the line by adjusting \(t\), making parametric equations a flexible and informational way to describe lines.