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. $$(1,1,1),(-10,-100,-1000)$$

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 Given Information

The problem gives a point on the line, \((1, 1, 1)\), and a direction vector parallel to the line, \((-10, -100, -1000)\). These represent a point through which the line passes and a vector that describes the direction of the line.

Step 2: Write Parametric Equations

Parametric equations for a line through a point \((x_0, y_0, z_0)\) with direction vector \((a, b, c)\) are given by: \[x = x_0 + at \y = y_0 + bt \z = z_0 + ct\] Thus, for this line:\[x = 1 - 10t,\]\[y = 1 - 100t,\]\[z = 1 - 1000t.\]

Step 3: Write the Symmetric Equations

Symmetric equations are obtained by solving the parametric equations for the parameter \(t\):\[t = \frac{x - 1}{-10} = \frac{y - 1}{-100} = \frac{z - 1}{-1000}\]These are the symmetric equations of the line.

Key Concepts

Symmetric equations

Symmetric equations provide a concise form to represent lines in three-dimensional space. They are derived from parametric equations and reveal the ratios of changes in each coordinate direction. Given the parametric equations:

• \( x = x_0 + at \)
• \( y = y_0 + bt \)
• \( z = z_0 + ct \)

The symmetric equations are obtained by isolating the parameter \( t \) in each equation.
This leads to:

• \( t = \frac{x - x_0}{a} \)
• \( = \frac{y - y_0}{b} \)
• \( = \frac{z - z_0}{c} \)

The line is expressed as a single equation by equating these expressions. In this exercise, the symmetric equations are:

• \( \frac{x - 1}{-10} = \frac{y - 1}{-100} = \frac{z - 1}{-1000} \)

This helps in visualizing how changes in one coordinate relate to changes in others.

Direction vector

The direction vector plays a crucial role in defining the orientation of a line in space. It is given in the form \((a, b, c)\), where each component indicates the direction in which the line extends.

If the direction vector is \((-10, -100, -1000)\), it indicates that the line moves predominantly in the negative directions of all axes. This is because all components are negative.

• The X-axis component, \(-10\), suggests that for each step along the parameter \( t \), the line moves 10 units negatively on the X-axis.
• Similarly, the Y-axis and Z-axis components (\(-100\) and \(-1000\), respectively) show a proportional decrease in those directions for each unit increase in \( t \).

Understanding the direction vector helps visualize the line's trajectory and ensures accurate line representation.

Point through which the line passes

To clearly describe a line in three-dimensional space, you need a reference point through which the line passes. This point is usually given or found and is denoted by its coordinates, \((x_0, y_0, z_0)\). In the problem, the point provided is \((1, 1, 1)\).

• This point acts as the starting point of the line and anchors it in space relative to other objects.
• Regardless of the direction and magnitude of the direction vector, the line will always traverse this point.

This point is crucial because, without it, we could not determine the specific location of the line, only its direction. Incorporating this point into our equations ensures we can fully describe the line's path in space.

Vector equation of a line

The vector equation of a line in space provides a compact way to represent all the points on the line using vectors. It is derived from the combination of a point on the line and a direction vector.
Such an equation is typically written as:
\[ \mathbf{r} = \mathbf{r}_0 + t\mathbf{v} \]where \( \mathbf{r}_0 \) is a position vector to a specific point on the line (like \((1,1,1)\) in this exercise), \( t \) is a scalar parameter, and \( \mathbf{v} \) is the direction vector (\((-10,-100,-1000)\)).
This equation reflects that every point \( \mathbf{r} \) on the line can be found by starting at \( \mathbf{r}_0 \) and moving \( t \) times along the direction given by \( \mathbf{v} \).

• Each different value of \( t \) provides a different point on the line.
• When \( t = 0 \), the point is exactly \( \mathbf{r}_0 \).
• Positive and negative values of \( t \) extend the line in opposite directions.

Understanding the vector equation helps see how the line extends infinitely in both directions in 3D space.