Problem 32 - Calculus III

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

The line through \(P\) and \(Q\) is represented by \(\mathbf{r}(t) = \langle 1 + 2t, 4 + 5t, -2 + 8t \rangle\).

Step 1: Understand the Vector-Valued Function

A vector-valued function describes a curve or path in space. For a line through two points \(P\) and \(Q\), you need to find the direction vector and then form the equation of the line.

Step 2: Find the Direction Vector

To find the direction vector of the line connecting points \(P(1, 4, -2)\) and \(Q(3, 9, 6)\), subtract the coordinates of \(P\) from the coordinates of \(Q\). So, the direction vector \(\mathbf{d}\) is \(\langle 3-1, 9-4, 6-(-2) \rangle = \langle 2, 5, 8 \rangle\).

Step 3: Formulate the Parametric Equations

Using point \(P(1, 4, -2)\) and direction vector \(\mathbf{d} = \langle 2, 5, 8 \rangle\), the parametric equations of the line are: \(x = 1 + 2t\), \(y = 4 + 5t\), \(z = -2 + 8t\), where \(t\) is a parameter.

Step 4: Define the Vector-Valued Function

The vector-valued function \(\mathbf{r}(t)\) for the line through points \(P\) and \(Q\) is \(\mathbf{r}(t) = \langle 1 + 2t, 4 + 5t, -2 + 8t \rangle\).

Step 5: Sketch the Line using a Graphing Utility

Input the vector function \(\mathbf{r}(t) = \langle 1 + 2t, 4 + 5t, -2 + 8t \rangle\) into a 3D graphing calculator. Adjust the parameter \(t\) to sufficiently large values (e.g., \(-10 \leq t \leq 10\)) to visualize the line passing through points \(P\) and \(Q\).

Key Concepts

Direction Vector

In the world of vector-valued functions, understanding the direction of a line is crucial. Imagine you want to describe the path between two points in a 3D space. You can do that with a vector called the **direction vector**.
To find the direction vector, identify the two points, say \( P(x_1, y_1, z_1) \) and \( Q(x_2, y_2, z_2) \). Then, subtract the coordinates of the starting point, \( P \), from the endpoint, \( Q \).
For example, for \( P(1, 4, -2) \) and \( Q(3, 9, 6) \), the direction vector \( \mathbf{d} \) is found as follows:

• \( x \)-component: \( 3 - 1 = 2 \)
• \( y \)-component: \( 9 - 4 = 5 \)
• \( z \)-component: \( 6 - (-2) = 8 \)

Thus, the direction vector \( \mathbf{d} \) is \( \langle 2, 5, 8 \rangle \). This vector tells you how much you move in each direction as you go from \( P \) to \( Q \).

Parametric Equations

Parametric equations describe how each coordinate of a point on a line is determined by a parameter, often denoted as \( t \). Through these equations, you can precisely describe a line in three dimensions.
To form parametric equations, you start with a point \( P(x_0, y_0, z_0) \) through which the line passes and a direction vector \( \mathbf{d} = \langle a, b, c \rangle \). The equations take the form:

• \( x = x_0 + at \)
• \( y = y_0 + bt \)
• \( z = z_0 + ct \)

Using our previous example with \( P(1, 4, -2) \) and direction vector \( \langle 2, 5, 8 \rangle \), the parametric equations become:\

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

These equations offer a straightforward method to describe each point on the line, as the parameter \( t \) varies.

3D Graphing

Graphing in three dimensions adds depth to our understanding of geometric shapes and paths. With modern graphing utilities, visualizing a line with parametric equations is straightforward and insightful.
Using the parametric equations \( x = 1 + 2t \), \( y = 4 + 5t \), and \( z = -2 + 8t \), you can graph the line in a 3D space. These utilities allow you to vary the parameter \( t \) over a range of values (e.g., \(-10 \leq t \leq 10\)) to fully visualize the line segment or an entire line.
When setting up your 3D graphing tool, input the equations and adjust the parameter range to ensure that many points on the line are plotted. This comprehensive visualization helps in understanding how the line extends in space and passes through the specified points.
Modern graphing calculators also provide interactive features, enabling you to rotate and zoom into the 3D graph for a better perspective of the line in the geometric space.