Problem 46 - Essential Calculus Early Transcendentals

Question

Which of the following four lines are parallel? Are any of them identical? $$L_{1} : x=1+6 t, \quad y=1-3 t, \quad z=12 t+5\( \)L_{2} : x=1+2 t, \quad y=t, \quad z=1+4 t\( \)L_{3} : 2 x-2=4-4 y=z+1\( \)L_{4} : \mathbf{r}=\langle 3,1,5\rangle+ t\langle 4,2,8\rangle$$

Step-by-Step Solution

Answer

Lines \( L_2 \) and \( L_4 \) are parallel; none are identical.

Step 1: Understanding the Line Equations

We need to compare the vectors that describe the direction of lines to determine if they are parallel or identical. Lines given in parametric form can be used to directly extract these direction vectors. For line equations given in other forms, convert them to parametric form first.

Step 2: Extract Direction Vectors

For line \( L_1 \), the direction vector is \( \langle 6, -3, 12 \rangle \). For line \( L_2 \), the direction vector is \( \langle 2, 1, 4 \rangle \). For line \( L_4 \), the direction vector is \( \langle 4, 2, 8 \rangle \).Now we convert line \( L_3: 2x - 2 = 4 - 4y = z + 1 \) to parametric form.

Step 3: Convert Line L3 to Parametric Form

Set \( 2x - 2 = 4 - 4y = z + 1 = k \).Solve for \( x, y, z \):1. \( 2x - 2 = k \Rightarrow x = \frac{k+2}{2} \)2. \( 4 - 4y = k \Rightarrow y = 1 - \frac{k}{4} \)3. \( z + 1 = k \Rightarrow z = k - 1 \)Direction vector for \( L_3 \) is \( \langle 1/2, -1/4, 1 \rangle \).

Step 4: Check for Parallel Lines

Lines are parallel if their direction vectors are scalar multiples of each other.- Compare \( \langle 6, -3, 12 \rangle \) with \( \langle 2, 1, 4 \rangle \): Not parallel, because their ratios are different.- Compare \( \langle 6, -3, 12 \rangle \) with \( \langle 4, 2, 8 \rangle \): Not parallel, because their ratios are different.- Compare \( \langle 6, -3, 12 \rangle \) with \( \langle 1/2, -1/4, 1 \rangle \): Not parallel, because their ratios are different.- Compare \( \langle 4, 2, 8 \rangle \) with \( \langle 2, 1, 4 \rangle \): Parallel, since \( \langle 4, 2, 8 \rangle = 2 \cdot \langle 2, 1, 4 \rangle \).

Step 5: Check for Identical Lines

Two lines are identical if they are parallel and share at least one point.- Check \( L_2 \) and \( L_4 \): - A point on \( L_2 \) is \( (1, 0, 1) \). - Check if this point satisfies \( L_4 \). - \( L_4 \) in parametric equations: \( x = 3 + 4t \), \( y = 1 + 2t \), \( z = 5 + 8t \). - Setting \( x = 1 \), \( y = 0 \), and \( z = 1 \) does not give a consistent solution for \( t \).Therefore, none of the lines are identical.

Key Concepts

Direction Vectors

Direction vectors play a critical role in understanding the orientation of lines in space. They help us determine if two lines are parallel. When you have a line described in parametric form like line \( L_1 : x=1+6t, y=1-3t, z=12t+5 \), the direction vector can be extracted from the coefficients of \( t \). Here, it would be \( \langle 6, -3, 12 \rangle \).

• This vector points in the direction along which the line extends.
• For parallelism, the direction vectors of two lines must be scalar multiples of each other.
• If two lines have the same direction vector, they are either parallel or identical.

For a line given in a different form, such as \( L_3: 2x - 2=4-4y=z+1 \), converting it to parametric form first will reveal the direction vector. In \( L_3 \), you solve equations to express \( x, y, z \) in forms that show the direction vector \( \langle 1/2, -1/4, 1 \rangle \). Thus, mastering direction vectors simplifies determining relationships between lines.

Parametric Equations

Parametric equations are a powerful way to describe lines in three-dimensional space. They express the position of any point on a line as a function of one or more parameters, typically denoted as \( t \). For example, in the line \( L_1: x=1+6t, y=1-3t, z=12t+5 \), each coordinate depends on the single parameter \( t \).

• Parametric equations allow for easy extraction of direction vectors.
• They also help in converting complex forms, like symmetrical or implicit equations, into a more usable format.
• By setting different values for \( t \), you can find any point on the line.

To convert lines in other forms such as \( L_3 \) discussed earlier, solve the equations step-by-step until each coordinate can be expressed in terms of a single parameter. This form makes it straightforward to analyze line properties, like finding a direction vector or checking for intersections.

Identical Lines

Two lines are considered identical if they not only run parallel but also share every point along their path. In mathematical terms, they must have identical direction vectors and a shared point. This concept is crucial when determining whether any given lines coincide.

• First, ensure the direction vectors are scalar multiples (i.e., they are parallel).
• Next, check if a point from one line satisfies the equations of the other.
• If both conditions are met, you have identical lines.

In the exercise, line \( L_2 \) and \( L_4 \) were checked for identicalness. They shared a direction vector relationship but failed to share a point, confirming that while they are parallel, they aren't identical. Understanding this distinction helps solve geometric problems involving lines in three-dimensional space.