Problem 19 - University Calculus: Early Transcendentals

Question

Find parametrizations for the line segments joining the points. Draw coordinate axes and sketch each segment, indicating the direction of increasing \(t\) for your parametrization. $$(2,0,2), \quad(0,2,0)$$

Step-by-Step Solution

Answer

The parametrization is \( x = 2 - 2t, y = 2t, z = 2 - 2t \) for \( 0 \leq t \leq 1 \).

Step 1: Determine the Direction Vector

To find the parametric equations of the line segment joining two points, we first need to determine the direction vector. The direction vector \( \vec{d} \) is obtained by subtracting the coordinates of the starting point \((2,0,2)\) from the coordinates of the ending point \((0,2,0)\). Thus, the direction vector is: \[ \vec{d} = (0, 2, 0) - (2, 0, 2) = (-2, 2, -2). \]

Step 2: Write the Parametric Equations

Using the starting point \((2,0,2)\) and the direction vector \( \vec{d} = (-2, 2, -2) \), we write the parametric equations for the line segment. Let \( t \) be the parameter such that \( 0 \leq t \leq 1 \). The parametric equations are: \[ \begin{align*} x(t) &= 2 - 2t, \ y(t) &= 0 + 2t, \ z(t) &= 2 - 2t. \end{align*} \]

Step 3: Sketch the Line Segment

First, draw a 3D coordinate system. The line segment starts at point \((2,0,2)\) and ends at point \((0,2,0)\). Plot these points. Starting at \((2,0,2)\), draw a straight line to \((0,2,0)\). Add an arrow on the line to indicate the direction of increasing \(t\), which moves from \((2,0,2)\) to \((0,2,0)\).

Key Concepts

Direction Vector

The direction vector is an essential component when working with line segments, especially in 3D spaces. This vector helps define the line's orientation by pointing from the starting to the ending point. To find it, you subtract the coordinates of the starting point from the ending point. For the given points

• starting point:
  
• ending point:

the direction vector is - -

direction vector
• 
  
• • 
    
  • direction vector
  to move as the parameter increases.

Parametric Equations

Parametric equations offer a neat way to express the coordinates of a point on a line as functions of a parameter, typically denoted as "t". For a line segment in 3D, each coordinate (x, y, z) is represented as a separate function of "t". Here's the general approach:

• Begin with a starting point, such as
  
• Use the direction vector, for instance, - -

To find the parametric equations:

• parametric equation

Thus, you get a system of parametric equations:- - - These equations help in understanding how the value of the parameter "t" affects the position along the line segment. It's important to note that for a segment, \(0 \leq t \leq 1\) ensures that the line doesn't extend beyond the specified ending points.

3D Coordinate System

Understanding the 3D coordinate system is crucial for visualizing how points and lines are positioned in space. In a 3D system, every point is defined by three coordinates:

• Coordinate axes
  
• the x-axis typically runs from left to right, the y-axis runs front to back or bottom to top, depending on the perspective, and the z-axis runs vertically.

The 3D coordinate system allows for comprehensive representation and movement within the geometrical realm. Visualizing a line segment starts with plotting its end points. For example, to draw the line segment connecting two points:

• 
  
• ,

you plot these points in the 3D space. The line segment between them represents movement across this three-dimensional space. To indicate motion from one point to another, arrows or directions can be added to show increasing or decreasing values of the parameter in parametric equations. Thus, the coordinate system not only helps in plotting points but also in visualizing their spatial relationships and movements.