Problem 32 - Calculus Volume 3

Question

Use a graphing utility to sketch each of the following vector-valued functions: The line through \(P\) and \(Q\) where \(P\) is (1,4,-2) and \(Q\) is (3,9,6)

Step-by-Step Solution

Answer

Graph the line using its parametric equations: \(x = 1 + 2t\), \(y = 4 + 5t\), \(z = -2 + 8t\).

Step 1: Find the Direction Vector

First, we find the direction vector for the line passing through the points \(P(1, 4, -2)\) and \(Q(3, 9, 6)\). The direction vector \(\mathbf{d}\) is given by the difference between point \(Q\) and point \(P\): \[\mathbf{d} = \begin{bmatrix} 3 - 1 \ 9 - 4 \ 6 - (-2) \end{bmatrix} = \begin{bmatrix} 2 \ 5 \ 8 \end{bmatrix}\]

Step 2: Write the Vector Equation

The vector equation of the line through \(P\) with direction \(\mathbf{d}\) is: \[\mathbf{r}(t) = \begin{bmatrix} 1 \ 4 \ -2 \end{bmatrix} + t \begin{bmatrix} 2 \ 5 \ 8 \end{bmatrix} = \begin{bmatrix} 1 + 2t \ 4 + 5t \ -2 + 8t \end{bmatrix}\] This represents the parametric form of the vector-valued function, where \(t\) is the parameter.

Step 3: Use a Graphing Utility

Open a graphing utility and enter the parametric equations derived in Step 2: - \(x = 1 + 2t\) - \(y = 4 + 5t\) - \(z = -2 + 8t\) Set the parameter \(t\) over an appropriate range, say \(-1 \leq t \leq 2\), and generate the plot to visualize the line in three-dimensional space.

Key Concepts

Direction Vector

When graphing a line through two points in space, a crucial step is determining the direction vector. This vector essentially describes the orientation of the line. To find the direction vector, subtract the coordinates of the initial point from the final point. For instance, if our points are \(P(1, 4, -2)\) and \(Q(3, 9, 6)\), the direction vector \(\mathbf{d}\) is calculated as:\[\mathbf{d} = \begin{bmatrix} 3 - 1 \ 9 - 4 \ 6 - (-2) \end{bmatrix} = \begin{bmatrix} 2 \ 5 \ 8 \end{bmatrix}\]This vector \(\begin{bmatrix} 2 \ 5 \ 8 \end{bmatrix}\) tells us that for every unit movement along the line, the x-coordinate increases by 2, the y-coordinate by 5, and the z-coordinate by 8. This information is essential for forming a precise mathematical model of the line.

Parametric Equations

Once the direction vector is established, we can construct parametric equations. These equations express each coordinate (x, y, z) of points on the line in terms of a single parameter, often denoted as \(t\). They describe a path traced by a moving point over a range of \(t\) values and are particularly useful for graphing.

• For the x-coordinate: \(x = 1 + 2t\)
• For the y-coordinate: \(y = 4 + 5t\)
• For the z-coordinate: \(z = -2 + 8t\)

Each equation shows how the respective coordinate changes as \(t\) varies. These equations provide a dynamic view of the line's path, representing not just a static set of points but a continuous line in 3D space.

Three-Dimensional Graphing

Graphing in three-dimensional space can effectively visualize vector-valued functions, like the line determined by two points. This process involves plotting a line by interpreting its parametric equations over a specified \(t\) range. Using a graphing utility, these equations allow you to see how the line extends through space.
Choose a \(t\) range that captures the line's intended length, for example, \(-1 \leq t \leq 2\). The resulting graph provides insight into how the line stretches and rotates in three dimensions. A functional grasp of 3D graphing is often vital for understanding complex mathematical and physical phenomena.

Vector Equation of a Line

The vector equation of a line unifies the direction vector and a point on the line to create a formula representing all line points. It combines a fixed starting point with a direction vector scaled by the parameter \(t\):\[\mathbf{r}(t) = \begin{bmatrix} 1 \ 4 \ -2 \end{bmatrix} + t \begin{bmatrix} 2 \ 5 \ 8 \end{bmatrix}\]This formulation highlights that any point on the line can be reached by starting at a known point, then "walking" along the direction vector a distance determined by \(t\).
The vector equation provides a concise and powerful way to deal with lines, especially when analyzing scenarios that involve intersections, angles, or projections in three-dimensional settings.