Problem 15 - Calculus Early Transcendentals

Question

\(15-18\) Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q .\) $$ P(0,0,0), \quad Q(1,2,3) $$

Step-by-Step Solution

Answer

Vector equation: \( \vec{r}(t) = (0,0,0) + t(1,2,3) \); Parametric: \( x=t, y=2t, z=3t \).

Step 1: Understand the Problem

We need to find the vector equation and parametric equations for the line segment from point \( P(0,0,0) \) to point \( Q(1,2,3) \). A line segment is described by a point and a direction vector.

Step 2: Determine the Direction Vector

The direction vector \( \vec{d} \) of the line can be found by subtracting the coordinates of \( P \) from \( Q \). So, \( \vec{d} = (1-0, 2-0, 3-0) = (1, 2, 3) \).

Step 3: Write the Vector Equation

The vector equation of a line can be written as \( \vec{r}(t) = \vec{P} + t \cdot \vec{d} \), where \( \vec{P} \) is the position vector of point \( P \), and \( \vec{d} \) is the direction vector. So, the equation is \( \vec{r}(t) = (0, 0, 0) + t(1, 2, 3) \).

Step 4: Establish Parameter Bounds

For a line segment, the parameter \( t \) varies between 0 and 1. This ensures that the segment starts at \( P \) when \( t = 0 \) and ends at \( Q \) when \( t = 1 \).

Step 5: Derive Parametric Equations

The parametric equations derived from the vector equation \( \vec{r}(t) = (0, 0, 0) + t(1, 2, 3) \) are: \( x(t) = 0 + t(1) = t \), \( y(t) = 0 + t(2) = 2t \), and \( z(t) = 0 + t(3) = 3t \). These are valid for \( t \) between 0 and 1.

Key Concepts

Parametric Equations

Parametric equations are a powerful way to represent a line or curve in space. They describe each coordinate (such as x, y, and z) as a function of a parameter, often denoted by \( t \). For a line, the parametric equations are derived from its vector equation.
The vector equation for the line segment joining two points, such as \( P(0,0,0) \) and \( Q(1,2,3) \), is \( \vec{r}(t) = (0, 0, 0) + t(1, 2, 3) \). From this, we can extract the parametric equations:

• \( x(t) = 0 + t \)
• \( y(t) = 0 + 2t \)
• \( z(t) = 0 + 3t \)

This means the x-coordinate is equal to \( t \), the y-coordinate is \( 2t \), and the z-coordinate is \( 3t \). These equations are used to determine positions along the line segment for different values of \( t \) within a specific range.

Direction Vector

The direction vector is a crucial component in defining lines, as it tells us the direction in which the line extends in space. To find the direction vector \( \vec{d} \) from point \( P \) to point \( Q \), we subtract the coordinates of \( P \) from those of \( Q \).
In this exercise, with points given as \( P(0,0,0) \) and \( Q(1,2,3) \), the direction vector is calculated as:
\[ \vec{d} = (1 - 0, 2 - 0, 3 - 0) = (1, 2, 3) \]
This vector precisely describes how far and in which direction the line extends from point \( P \). The components of the direction vector (1, 2, 3) indicate that for every 1 unit the line extends in the x-direction, it extends 2 units in the y-direction and 3 units in the z-direction.

Line Segment

A line segment, unlike an infinite line, has defined endpoints, making it a finite portion of a line. In vector notation, we represent a line segment using a point and a direction vector, along with parameter bounds.
For our line segment from point \( P(0,0,0) \) to \( Q(1,2,3) \), it is defined using the point \( P \), the direction vector \( \vec{d} = (1, 2, 3) \), and bounds on the parameter \( t \) (which will be discussed further).

• The start of the segment is represented by \( \vec{r}(0) = (0, 0, 0) \)
• The end is denoted by \( \vec{r}(1) = (1, 2, 3) \)

Line segments are immensely useful in geometry and physics because they help to model finite distances between points in space.

Parameter Bounds

Parameter bounds are what define the start and end of a line segment. They are the values of \( t \), the parameter, that determine where the line segment begins and where it ends.
For our specific problem, the bounds for \( t \) are 0 and 1. These bounds ensure that the segment starts at \( P(0,0,0) \) when \( t = 0 \) and ends at \( Q(1,2,3) \) when \( t = 1 \).
This means the parameter \( t \) captures every point on the line segment, traveling in the direction of the vector \( \vec{d} \). Each possible value of \( t \) within these bounds corresponds to a unique point on the segment: