Problem 18 - Essential Calculus Early Transcendentals

Question

\(17-20=\) Determine whether the lines \(L_{1}\) and \(L_{2}\) are parallel, skew, or intersecting. If they intersect, find the point of intersection. $$ \begin{array}{l}{L_{1 :} x=5-12 t, \quad y=3+9 t, \quad z=1-3 t} \\ {L_{2} : x=3+8 s, \quad y=-6 s, \quad z=7+2 s}\end{array} $$

Step-by-Step Solution

Answer

The lines \(L_1\) and \(L_2\) are parallel.

Step 1: Understand the Line Equations

Lines are given in the parametric form. \(L_{1}\) is given by the equations \(x = 5 - 12t\), \(y = 3 + 9t\), \(z = 1 - 3t\). \(L_{2}\) is given by the equations \(x = 3 + 8s\), \(y = -6s\), \(z = 7 + 2s\). Our task is to determine the relationship between these two lines.

Step 2: Identify Direction Vectors

From the parametric equations, identify the direction vectors of the lines. For \(L_{1}\), the direction vector is \(\langle -12, 9, -3 \rangle\). For \(L_{2}\), the direction vector is \(\langle 8, -6, 2 \rangle\).

Step 3: Check if Lines are Parallel

To be parallel, the direction vectors should be scalar multiples of each other. Check if \(\frac{-12}{8} = \frac{9}{-6} = \frac{-3}{2}\). Calculate these ratios: \(\frac{-12}{8} = -1.5\), \(\frac{9}{-6} = -1.5\), and \(\frac{-3}{2} = -1.5\). Since the ratios are the same, the lines are parallel.

Key Concepts

Parametric Equations

Parametric equations are a way to express the coordinates of the points that make up geometric objects like lines using one or more parameters. In this case, parameters are often represented as "t" and "s" for lines \( L_1 \) and \( L_2 \) respectively. Parametric equations allow you to translate a more abstract geometric concept into a concrete format that can be worked with mathematically.
For line \( L_1 \), the parametric equations are given as:

• \( x = 5 - 12t \)
• \( y = 3 + 9t \)
• \( z = 1 - 3t \)

Similarly, for line \( L_2 \), the equations are:

• \( x = 3 + 8s \)
• \( y = -6s \)
• \( z = 7 + 2s \)

In these equations, "t" and "s" can take any real number value, which means both lines extend infinitely in both directions. By substituting different values of "t" or "s", you can trace out all the points that lie on a line. Each variable (x, y, and z) corresponds to a spatial dimension, so together, they locate a point in 3D space.

Direction Vectors

A direction vector indicates the direction in which a line extends and is derived directly from the coefficients of the parametric equations. It essentially serves as an arrow pointing down the line's path.
For line \( L_1 \), the direction vector is \( \langle -12, 9, -3 \rangle \). This is found by taking the coefficients of parameter "t" from each component equation.
For line \( L_2 \), the direction vector is \( \langle 8, -6, 2 \rangle \) and is obtained similarly from the "s" parameter. These vectors can help determine several properties of the lines, such as whether they are parallel or if they may intersect.
Direction vectors act like a GPS device for the line, offering guidance on how the line moves through space. If two direction vectors are in the same or exactly opposite directions, the lines they describe are parallel.

Scalar Multiples

Determining if two lines are parallel involves checking if their direction vectors are scalar multiples of each other. A scalar multiple occurs when the components of one vector can be obtained by multiplying each component of the other vector by the same number (called a scalar).
To check if the direction vectors \( \langle -12, 9, -3 \rangle \) and \( \langle 8, -6, 2 \rangle \) are scalar multiples, you compare each corresponding component:

• \( \frac{-12}{8} = -1.5 \)
• \( \frac{9}{-6} = -1.5 \)
• \( \frac{-3}{2} = -1.5 \)

All these ratios are equal, confirming that the lines are indeed parallel. This means they will never meet and continue alongside each other indefinitely. If one or more of these ratios differed, it would imply that the lines might not be parallel and could possibly intersect somewhere in the infinite expanse of space.