Problem 14 - Precalculus Mathematics for Calculus

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(t) = 12\\), \\(y(t) = 16 - 22t\\), \\(z(t) = 18 - 18t\\).

Step 1: Determine the Direction Vector

To find the parametric equations of a line, we first need to find the direction vector, which can be obtained by subtracting the coordinates of point \(P\) from point \(Q\). The direction vector \(\mathbf{v}\) is given by \((Q_x - P_x, Q_y - P_y, Q_z - P_z)\). Thus, \ \mathbf{v} = (12-12, -6-16, 0-18) = (0, -22, -18) \.

Step 2: Write the Parametric Equations of the Line

The parametric equations of a line can be formulated using a point on the line and the direction vector. Using point \(P(12, 16, 18)\) as the point, and \(\mathbf{v} = (0, -22, -18)\) as the direction vector, the parametric equations are: 1. \(x(t) = 12\) 2. \(y(t) = 16 - 22t\) 3. \(z(t) = 18 - 18t\).

Key Concepts

direction vector

When dealing with lines in three-dimensional space, a direction vector is crucial. This vector provides the necessary orientation of the line. It indicates the path that the line follows. To calculate the direction vector between two points, such as points \(P\) and \(Q\), you subtract the coordinates of the starting point (in this case, \(P\)) from the coordinates of the endpoint (\(Q\)).
For our example, the direction vector \(\mathbf{v}\) came out as \((0, -22, -18)\). Each component of this vector reflects the change in the x, y, and z directions respectively. Here:

• The x-component (0) indicates there is no movement along the x-axis.
• The y-component (-22) suggests a downward movement along the y-axis.
• The z-component (-18) shows a downward shift along the z-axis.

Direction vectors are essential in many applications, from physics to computer graphics, as they illustrate how a line progresses through space.

3D coordinate geometry

3D coordinate geometry extends the concepts of two-dimensional geometry into three dimensions, allowing us to locate any point in space with an ordered triple of numbers, often denoted as \((x, y, z)\). When working in 3D space, like with our exercise, each point is expressed through its position along three perpendicular axes: x, y, and z.
The line examined in the exercise using points \(P(12, 16, 18)\) and \(Q(12, -6, 0)\) is a example of connecting these concepts. Geometry in three dimensions not only helps in visualizing the positions and traces of lines but also plays a critical role in fields like engineering and architecture where spatial reasoning is paramount.
Applications within 3D coordinate geometry also include calculating distances between points, finding intersections of lines, and understanding object rotations and translations. By grasping 3D geometry basics, you will be able to tackle more complex spatial problems effectively.

lines in space

A line in three-dimensional space is a set of points extending without end in opposite directions. This profound concept, however, can be represented quite simply with parametric equations. These equations clarify how each point on the line can be described based on a single parameter, typically denoted \(t\). In our exercise, defining the parametric equations involved using a point on the line, point \(P(12, 16, 18)\), and a direction vector \(\mathbf{v} = (0, -22, -18)\).
The parametric equations arrived at are:

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

Each of these equations reveals how the coordinates of any point on the line vary with \(t\). For \(x(t)\), no change occurs as \(t\) changes, indicating constant x-value across the line. However, both \(y(t)\) and \(z(t)\) are functions of \(t\) reflecting movement along their respective axes.
Understanding how lines in space work is not only key in mathematics but is foundational in fields involving spatial data like physics and engineering mechanics.