Problem 7 - 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. \((1,1,1),\langle-10,-100,-1000\rangle\)

Step-by-Step Solution

Answer

Parametric: \(x = 1 - 10t\), \(y = 1 - 100t\), \(z = 1 - 1000t\). Symmetric: \(\frac{x-1}{-10} = \frac{y-1}{-100} = \frac{z-1}{-1000}\).

Step 1: Understand the Components

The line must go through the point \((1,1,1)\) and is parallel to the vector \\(\langle-10,-100,-1000\rangle\). This vector gives the direction of the line.

Step 2: Write Parametric Equations

The parametric equations of a line can be expressed as \( x = x_0 + at \), \( y = y_0 + bt \), \( z = z_0 + ct \), where \((x_0, y_0, z_0)\) is the point and \(\langle a, b, c \rangle\) is the direction vector. For our case, substitute in \((1,1,1)\) and \(\langle-10, -100, -1000\rangle\) as follows: \[ \begin{align*} k &= 1 - 10t, \y &= 1 - 100t,\ z &= 1 - 1000t \end{align*} \] These are the parametric equations of the line.

Step 3: Write Symmetric Equations

The symmetric equations are derived from the parametric equations by setting the parameter \( t \) equal in each of them: \[ \frac{x - 1}{-10} = \frac{y - 1}{-100} = \frac{z - 1}{-1000}. \] These are the symmetric equations for the line.

Key Concepts

Symmetric Equations

Symmetric equations are a way to describe a line in three-dimensional space without the parameter \( t \). They stem from parametric equations by eliminating \( t \) and equating each expression for the three coordinates. For a line passing through a point \((x_0, y_0, z_0)\) with a direction vector \(\langle a, b, c \rangle\):

• The parametric equations are: \( x = x_0 + at \), \( y = y_0 + bt \), and \( z = z_0 + ct \).
• To obtain the symmetric equations, set all terms equal to the parameter \( t \): \( \frac{x - x_0}{a} = \frac{y - y_0}{b} = \frac{z - z_0}{c} \).

The result is a scalar multiple equation that relates the increments along each axis to the components of the direction vector. This formulation helps visualize the line's path through space.

Direction Vector

A direction vector is vital in defining a line in space. This vector provides insight into the line's orientation by indicating its direction and slope.

The vector is usually denoted as \( \langle a, b, c \rangle \), where:

• \( a \), \( b \), and \( c \) represent the changes in the \( x \), \( y \), and \( z \) coordinates respectively.
• It shows in which directions and how fast the line moves in respect to each of the coordinate axes.
• Moreover, scaling the direction vector does not change the line's path; it only affects the speed at which the line is traced.

The direction vector is essential in both parametric and symmetric equations, allowing for the line to be represented effectively in mathematical terms.

Line in 3D

Describing a line in three-dimensional space can be more complex than in two dimensions. In three dimensions, a line is determined by a point it passes through and a direction vector indicating its path.

To represent a line in 3D:

• Start with a known point \((x_0, y_0, z_0)\) on the line.
• Use a direction vector \( \langle a, b, c \rangle \) describing the line's orientation.
• The parametric equations \( x = x_0 + at \), \( y = y_0 + bt \), and \( z = z_0 + ct \) give a point-specific formulation, enabling precise tracing of the line's path for varying values of \( t \).
• Alternatively, the symmetric equations reveal the inherent relationships amongst points along the line without relying on the parameter \( t \).

Understanding these various formats for defining a line enhances spatial reasoning and flexibility in solving geometric problems.

Parametric Formulation

Parametric equations are a powerful way to represent lines, especially in three dimensions. They define a line using a parameter such as \( t \), creating distinct equations for each spatial dimension. This method breaks down the geometric path into a series of coordinate-aligned steps.

Key components in parametric formulation are:

• A base point \((x_0, y_0, z_0)\) which is a specific point the line passes through.
• A direction vector \( \langle a, b, c \rangle \) that states how the line progresses from the base point.

In the context of the exercise given, the parametric equations were derived using starting point \((1,1,1)\) and direction vector \(\langle -10, -100, -1000 \rangle\). They take the form:
\[ \begin{align*} x &= 1 - 10t, \ y &= 1 - 100t, \ z &= 1 - 1000t \end{align*} \]
This allows any point on the line to be calculated by entering a specific value of \( t \), showcasing its position in 3D space.