Problem 17 - Calculus Early Transcendentals

Question

\(15-18\) Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q .\) $$ P(1,-1,2), \quad Q(4,1,7) $$

Step-by-Step Solution

Answer

Vector equation: \( \mathbf{r}(t) = (1, -1, 2) + t(3, 2, 5) \), Parametric: \( x=1+3t \), \( y=-1+2t \), \( z=2+5t \).

Step 1: Determine the Direction Vector

To find the direction vector of the line segment, subtract the coordinates of point \( P \) from point \( Q \). The direction vector \( \mathbf{d} \) is given by: \( \mathbf{d} = Q - P = (4-1, 1-(-1), 7-2) = (3, 2, 5) \).

Step 2: Write the Vector Equation

The vector equation for the line segment can be expressed as \( \mathbf{r}(t) = \mathbf{P} + t(\mathbf{d}) \), where \( t \) is a parameter varying from 0 to 1. Substituting \( \mathbf{P} = (1, -1, 2) \) and \( \mathbf{d} = (3, 2, 5) \), the vector equation is: \( \mathbf{r}(t) = (1, -1, 2) + t(3, 2, 5) \).

Step 3: Derive Parametric Equations

To obtain the parametric equations from the vector equation, express each coordinate as a function of \( t \). Therefore, \( x(t) = 1 + 3t \), \( y(t) = -1 + 2t \), and \( z(t) = 2 + 5t \). These are the parametric equations of the line segment.

Key Concepts

Vector Equation

A vector equation represents a line by combining a fixed position and a direction vector. It forms the backbone of describing lines in space. In the context of this exercise, the line segment joining points \(P\) and \(Q\) is defined using a vector equation. Every point on the line can be described as \(\mathbf{r}(t) = \mathbf{P} + t(\mathbf{d})\). This equation incorporates the starting point \(\mathbf{P}\), which is the base location on the line, and the direction vector \(\mathbf{d}\), which guides the path of the line. The variable \(t\) controls how far along the segment the point is, ranging from 0 to 1. The vector equation provides a compact way to express the line's path, making it easy to calculate any specific point on the segment.

Parametric Equations

Parametric equations break down the vector equation into separate parts for each coordinate. In essence, they express each spatial dimension as a distinct function of \(t\). For our line connecting \(P\) and \(Q\), the parametric equations derived from the vector equation are:

• \( x(t) = 1 + 3t \)
• \( y(t) = -1 + 2t \)
• \( z(t) = 2 + 5t \)

These equations tell us how the x, y, and z coordinates change as you move along the line segment from \(P\) to \(Q\). By adjusting \(t\) between 0 and 1, you can determine any point on this line. This parametric form is especially useful in applications that require modeling motion or paths in three-dimensional space.

Direction Vector

The direction vector is crucial for understanding the orientation and heading of a line. It arises from the difference between the coordinates of two points on the line. For points \(P(1, -1, 2)\) and \(Q(4, 1, 7)\), the direction vector \(\mathbf{d}\) is calculated by subtracting the coordinates of \(P\) from those of \(Q\). Thus, we have \(\mathbf{d} = (4 - 1, 1 - (-1), 7 - 2) = (3, 2, 5)\). This vector indicates how the line shifts in the x, y, and z directions, respectively. It essentially outlines the step pattern of the line, determining both direction and scale without asserting a specific start or end. The direction vector serves as an arrow pointing through the entire line segment.

Coordinate Geometry

Coordinate geometry is the study of geometry using a coordinate system, which integrates algebraic methods to solve geometrical problems. It provides a framework to describe shapes and lines using equations and coordinates. In this exercise, by employing coordinate geometry, we created a means to precisely describe the path between points \(P\) and \(Q\) in three dimensions. This includes distinguishing defined parts such as points \(P\) and \(Q\), the line through these points, and the methodology to find a path between them using equations. By using coordinate geometry, especially in three-dimensional space, one gains the ability to visualize and dissect complex geometric problems with ease, thereby transforming abstract concepts into calculable and visualized solutions.