Problem 13 - Essential Calculus Early Transcendentals

Question

\(13-16=\) Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q .\) $$P(2,0,0), \quad Q(6,2,-2)$$

Step-by-Step Solution

Answer

The vector equation is \( \mathbf{r}(t) = (2,0,0) + t(4,2,-2) \) and the parametric equations are \( x=2+4t, y=2t, z=-2t \) for \( 0 \leq t \leq 1 \).

Step 1: Identify the Direction Vector

To find the direction vector for the line, we subtract the coordinates of point \(P\) from point \(Q\). The direction vector \( \mathbf{d} \) is given by: \[ \mathbf{d} = Q - P = (6-2, 2-0, -2-0) = (4, 2, -2). \]

Step 2: Write the Vector Equation

The vector equation of the line segment can be expressed using the position vector of point \(P\) and the direction vector \(\mathbf{d}\). The vector equation is: \[ \mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{d}, \] where \(\mathbf{r}_0 = (2, 0, 0)\) and \(\mathbf{d} = (4, 2, -2)\). Thus, \[ \mathbf{r}(t) = (2, 0, 0) + t(4, 2, -2). \]

Step 3: Determine the Range of Parameter \(t\)

Since we want the equation for the line segment between points \(P\) and \(Q\), \(t\) should range from 0 to 1 inclusive. For \(t=0\), the position is at \(P\) and for \(t=1\), the position is at \(Q\).

Step 4: Write the Parametric Equations

From the vector equation, we can derive the parametric equations. The vector equation is \( \mathbf{r}(t) = (2, 0, 0) + t(4, 2, -2) \). Splitting this into parametric equations, we get: \[ x = 2 + 4t, \] \[ y = 0 + 2t, \] \[ z = 0 - 2t. \] These equations are valid for \(0 \leq t \leq 1\).

Key Concepts

Direction Vector

In the world of vectors, a direction vector is essential when describing a line or a line segment. It's like a guide that points you in the right direction from one point to another. To find the direction vector between two points, you subtract the coordinates of the starting point from the ending point. Think of this as finding the difference in each dimension. This process transforms your two fixed points, in this case, points \(P\) and \(Q\), into a single vector that essentially "points" from \(P\) to \(Q\).

• Start with point \(P(2, 0, 0)\) and point \(Q(6, 2, -2)\).
• The direction vector \( \mathbf{d} \) is calculated as \( Q - P \).
• This gives us \( \mathbf{d} = (6-2, 2-0, -2-0) = (4, 2, -2) \).

With this direction vector, you now have a foundational piece needed to describe the line segment joining your two points.

Parametric Equations

Parametric equations use a parameter, typically denoted as \(t\), to express the coordinates of the points on a line or line segment. These equations allow us to express the line in terms of simpler algebraic terms, which are easier to handle and understand.From a vector equation, which we write as \( \mathbf{r}(t) = \mathbf{r}_0 + t \mathbf{d} \), we can get parametric equations for each coordinate. Here:

• \(\mathbf{r}_0 = (2, 0, 0)\) is our initial position vector.
• \(\mathbf{d} = (4, 2, -2)\) is our direction vector.
• Our vector equation becomes \(\mathbf{r}(t) = (2, 0, 0) + t(4, 2, -2)\).

Breaking it down:

• \(x = 2 + 4t\)
• \(y = 0 + 2t\)
• \(z = 0 - 2t\)

These equations describe how the coordinates \( (x, y, z) \) change as \(t\) varies from 0 to 1. Each value of \(t\) gives you a specific point on the line segment linking \(P\) and \(Q\). This makes it easy to find any point you might need on the segment.

Line Segment

A line segment is the portion of a line that lies between two endpoints. It's like a piece of spaghetti stretch between two specific points in space. Unlike an infinite line, a line segment stops at its endpoints; it does not go on forever.The concept of a line segment is important when we're working with vector equations and parametric equations, especially since they allow us to pinpoint exactly where the line starts and ends. In our scenario with points \(P\) and \(Q\):

• The segment begins at point \(P(2, 0, 0)\) and ends at point \(Q(6, 2, -2)\).
• The parameter \(t\) restricts the equation, ranging from 0 to 1.
• When \(t=0\), the position vector is \(P\) and when \(t=1\), the vector reaches \(Q\).

This understanding helps us ensure we're only finding the points that are truly on the segment between \(P\) and \(Q\), providing a clear and finite path between them.