Problem 18 - Calculus Early Transcendentals

Question

\(15-18\) Find a vector equation and parametric equations for the line segment that joins \(P\) to \(Q .\) $$ P(-2,4,0), \quad Q(6,-1,2) $$

Step-by-Step Solution

Answer

The vector equation is \( \vec{r}(t) = (-2, 4, 0) + t(8, -5, 2) \) and parametric equations are \( x = -2 + 8t, y = 4 - 5t, z = 2t \) for \( 0 \leq t \leq 1 \).

Step 1: Determine the Direction Vector

First, we need to find the direction vector from point \( P \) to point \( Q \). This is done by subtracting the coordinates of \( P \) from \( Q \). The direction vector \( \vec{d} \) is \( Q - P = (6 - (-2), -1 - 4, 2 - 0) = (8, -5, 2).\)

Step 2: Establish the Vector Equation of the Line

To find the vector equation of the line segment, we use the formula \( \vec{r}(t) = \vec{P} + t \cdot \vec{d} \) where \( \vec{P} \) is the position vector of point \( P \) and \( t \) is a parameter. The vector equation becomes \( \vec{r}(t) = (-2, 4, 0) + t(8, -5, 2).\)

Step 3: Write the Parametric Equations

Extract the components from the vector equation to write the parametric equations. From \( \vec{r}(t) = (-2, 4, 0) + t(8, -5, 2) \), the parametric equations are: \[ x = -2 + 8t, \]\[ y = 4 - 5t, \]\[ z = 2t. \]

Step 4: Specify the Parameter Range

Since the question asks for the line segment, the parameter \( t \) should vary between 0 and 1 to ensure the segment spans from \( P \) to \( Q \). Thus, the parameter range is \( 0 \leq t \leq 1 \).

Key Concepts

Parametric Equations

Parametric equations represent a way to describe a line or curve in space where each coordinate depends on a common parameter, usually denoted as \( t \). This makes it easier to define the position of points along a line or curve.

• Each coordinate in space (\( x \), \( y \), and \( z \) for three dimensions) is expressed as a function of \( t \).
• This helps in representing not just infinite lines, but also specific segments where the parameter \( t \) is restricted within certain values.

For example, the parametric equations \( x = -2 + 8t \), \( y = 4 - 5t \), and \( z = 2t \) represent the line segment from one point to another when \( t \) moves from 0 to 1.

Direction Vector

The direction vector is crucial in defining the orientation and direction of a line in space.

• It tells us how the line extends from one point to another by highlighting the difference between the two points.
• To find the direction vector \( \vec{d} \) between two points \( P(-2,4,0) \) and \( Q(6,-1,2) \), you perform a simple subtraction: \( \vec{d} = Q - P = (6 - (-2), -1 - 4, 2 - 0) = (8, -5, 2) \).

The resulting vector \( (8, -5, 2) \) not only indicates direction but also the magnitude of the shift from \( P \) to \( Q \).
This vector becomes part of the equation used to describe the line or segment in space.

Line Segment

A line segment is simply a part of a line that connects two points, without extending beyond them.

• It’s different from an infinite line as it has clear starting and ending points. In our case, these points are \( P(-2,4,0) \) and \( Q(6,-1,2) \).
• To describe this segment analytically, we use the derived parametric equations and restrict the parameter \( t \) such that it spans the desired endpoints.

For the line segment from \( P \) to \( Q \), \( t \) varies from 0 to 1, ensuring the segment is fully described.
This restrained parameter reflects the finite nature of a line segment.

Coordinate Geometry

Coordinate geometry, or analytic geometry, combines algebra and geometry to solve problems involving coordinates in space.

• It provides a powerful framework to describe geometric shapes and forms, like lines, circles, and polygons, using algebraic equations.
• For lines and segments, it employs coordinates to express points and direction, allowing precise and analytical study of geometric problems.

The entire process of finding vector and parametric equations of a line or segment between two points is a clear application of coordinate geometry.
This field helps visualize and calculate geometric entities in two, three, or even higher-dimensional spaces.