Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

Write both the parametric equations and the symmetric equations for the line through the given point parallel to the given vector. $$(4,5,6),\langle 3,2,1\rangle$$

Step-by-Step Solution

Answer

Parametric: \(x = 4 + 3t, y = 5 + 2t, z = 6 + t\). Symmetric: \(\frac{x-4}{3} = \frac{y-5}{2} = z-6\).

Step 1: Identify Point and Direction Vector

The given point through which the line passes is \((4,5,6)\). The direction vector for the line is \(\langle 3,2,1 \rangle\). These will be used to form both parametric and symmetric equations.

Step 2: Write Parametric Equations

The parametric equations express the line using the point \((4,5,6)\) and the direction vector \(\langle 3,2,1 \rangle\). They are given by: \(x = 4 + 3t\), \(y = 5 + 2t\), and \(z = 6 + 1t\), where \(t\) is a parameter.

Step 3: Eliminate Parameter for Symmetric Equations

To find the symmetric equations, solve each parametric equation for the parameter \(t\). From \(x = 4 + 3t\), we have \(t = \frac{x-4}{3}\). From \(y = 5 + 2t\), we have \(t = \frac{y-5}{2}\). From \(z = 6 + t\), we have \(t = z-6\).

Step 4: Write Symmetric Equations

Set the equations from the previous step equal to each other to eliminate the parameter: \(\frac{x-4}{3} = \frac{y-5}{2} = z-6\). These are the symmetric equations of the line.

Key Concepts

Symmetric Equations

Symmetric equations are a type of line equation that conveniently express a line in three-dimensional space without directly using a parameter. To derive symmetric equations, you use the parametric equations of a line. These parametric equations express coordinates as functions of a parameter, usually denoted by \( t \).
First, solve each parametric equation for \( t \). You'll get three separate expressions for \( t \), one from each coordinate.
Next, set these expressions equal to each other. In this example:

• For \( x \), \( t = \frac{x-4}{3} \)
• For \( y \), \( t = \frac{y-5}{2} \)
• For \( z \), \( t = z-6 \)

Equating these gives the symmetric form: \( \frac{x-4}{3} = \frac{y-5}{2} = z-6 \).
The symmetric equation represents the same line as the parametric form but does not involve parameters explicitly.

Direction Vector

The direction vector plays a crucial role in defining the line in 3D space. It's essentially a vector that points in the direction the line extends. In this scenario, the direction vector is \( \langle 3,2,1 \rangle \).
A direction vector indicates how much the line moves in the x, y, and z directions as you travel along the line.
Here's why it matters:

• The components of this vector become the coefficients of \( t \) in your parametric equations.
• In symmetric equations, these components are used as divisors.

This means the line is described in the context of its vector-wise travel from a point, showcasing its incline or decline in space.

Line Equation

A line in 3D space can be described by both parametric and symmetric equations. The parametric equations can be written given a point on the line and a direction vector.
For example, if a line passes through the point \( (4,5,6) \) and is parallel to the vector \( \langle 3,2,1 \rangle \), its parametric equations are:

• \( x = 4 + 3t \)
• \( y = 5 + 2t \)
• \( z = 6 + 1t \)

These equations show the position of a point on the line as \( t \) varies.
Symmetric equations, however, remove \( t \) and focus entirely on the coordinates. They can often be more intuitive when comparing or combining lines.

3D Coordinate Geometry

In 3D coordinate geometry, lines, planes, and volumes are defined by their positions and orientations in a three-dimensional space. Understanding these concepts is essential for solving problems involving spatial relationships.
Key concepts include:

• Points: Defined by coordinates \( (x, y, z) \).
• Vectors: Represented as \( \langle a, b, c \rangle \), indicating direction and magnitude.
• Equations: Describe lines and surfaces, with both parametric and symmetric forms offering unique insights.

Mastering these allows for the calculation and visualization of lines like the one in the exercise, aiding in deeper spatial comprehension.