Problem 16 - Calculus Early Transcendentals

Question

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

Step-by-Step Solution

Answer

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

Step 1: Define the Vector OP

To find a vector equation of the line segment from point \(P\) to point \(Q\), we first need a vector that represents point \(P\). This is given by vector \(\overrightarrow{OP} = \langle 1, 0, 1 \rangle\).

Step 2: Define the Vector PQ

Next, we find the vector from \(P\) to \(Q\). This vector, \(\overrightarrow{PQ}\), is obtained by subtracting the coordinates of \(P\) from those of \(Q\). Thus, \(\overrightarrow{PQ} = Q - P = \langle 2, 3, 1 \rangle - \langle 1, 0, 1 \rangle = \langle 1, 3, 0 \rangle\).

Step 3: Write the Vector Equation for the Line

The vector equation for the line segment can now be written as \(\vec{r}(t) = \overrightarrow{OP} + t\cdot\overrightarrow{PQ}\), where \(t\) ranges from 0 to 1. This gives us \(\vec{r}(t) = \langle 1, 0, 1 \rangle + t\langle 1, 3, 0 \rangle\).

Step 4: Derive the Parametric Equations

Finally, we can convert the vector equation into parametric equations. By distributing \(t\) and adding the components, we get: - \(x(t) = 1 + t \cdot 1 = 1 + t\) - \(y(t) = 0 + t \cdot 3 = 3t\) - \(z(t) = 1 + t \cdot 0 = 1\) Thus, the parametric equations are \(x = 1 + t\), \(y = 3t\), \(z = 1\), where \(0 \leq t \leq 1\).

Key Concepts

Parametric Equations

Parametric equations are a way to express a set of equations such that one or more parameters define the coordinates of a point. In the case of a line, these parameters help us track a point as it moves along the line. An essential part of understanding parametric equations involves recognizing that they provide a compact representation of geometric paths. For a line segment, the parameter typically ranges between 0 and 1, which helps us plot points from the start to the end of the segment. In our exercise, we have the line segment from point \(P(1,0,1)\) to point \(Q(2,3,1)\). To find the parametric equations, we can use the vector equation: \[ \vec{r}(t) = \langle 1, 0, 1 \rangle + t \langle 1, 3, 0 \rangle \] Here, each component involves the parameter \(t\), resulting in:

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

These equations show how to progress from the point \(P\) to \(Q\) by varying the parameter \(t\). For our segment, \(0 \leq t \leq 1\) guarantees that we only cover the points between \(P\) and \(Q\).

Vector Algebra

Vector algebra is a branch of mathematics that involves vectors, or quantities that have both magnitude and direction. Vectors are fundamental in representing physical quantities like force or velocity. In our problem, vector algebra helps us define and manipulate vectors to describe a line segment. First, we identify the position vector \(\overrightarrow{OP} = \langle 1, 0, 1 \rangle\). This vector tells us where point \(P\) is located. Next, we calculate the direction vector \(\overrightarrow{PQ}\) by subtracting \(P\)'s coordinates from \(Q\)'s, giving us \(\langle 1, 3, 0 \rangle\). The importance of vector algebra here is in constructing a vector equation for the line segment:

• \(\vec{r}(t) = \overrightarrow{OP} + t \cdot \overrightarrow{PQ}\)
• Here, \(t\) is a scalar parameter that scales the direction vector \(\overrightarrow{PQ}\).

By breaking down vectors and exploring components, vector algebra gives us the tools needed to express, manipulate, and understand points and lines in space.

Line Segments

A line segment is a part of a line that has two endpoints and covers all the points in between. In mathematics and geometry, a line segment is defined in space or on a plane. This concept differs from a line, which extends infinitely in both directions. Understanding line segments is crucial, especially when working with vectors and parametric equations. In our scenario, the aim is to capture the line segment that extends from point \(P\) to point \(Q\). The vector equation \(\vec{r}(t) = \langle 1, 0, 1 \rangle + t\langle 1, 3, 0 \rangle\) allows us to express each point along the segment. By choosing \(t\) values between 0 and 1, we ensure that only points on the segment from \(P\) to \(Q\) are included. Line segments are important because:

• They have a definite length, given by the distance between their endpoints.
• They are often used in computer graphics, animation, and various engineering fields for precise modeling.

Whether in two-dimensional or three-dimensional space, line segments provide valuable insight into spatial relationships and are foundational in geometry and vector applications.