Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

Let \(P\) and \(Q\) be points on nonintersecting skew lines with directions \(\mathbf{n}_{1}\) and \(\mathbf{n}_{2}\), and let \(\mathbf{n}=\mathbf{n}_{1} \times \mathbf{n}_{2}\) (Figure 8). Show that the distance \(d\) between these lines is given by $$ d=\frac{|\overrightarrow{P Q} \cdot \mathbf{n}|}{\|\mathbf{n}\|} $$ and use this result to find the distance between each pair of lines in parts (a) and (b). (a) \(\frac{x-3}{1}=\frac{y+2}{1}=\frac{z-1}{2}\) and \(\frac{x+4}{3}=\frac{y+5}{4}=\frac{z}{5}\) (b) \(x=1+2 t, y=-2+3 t, z=-4 t\) and \(x=3 t, y=1+t\), \(z=-5 t\)

Step-by-Step Solution

Answer

Use the formula \( d = \frac{|\overrightarrow{PQ} \cdot \mathbf{n}|}{\|\mathbf{n}\|} \) to calculate the distance for each pair of skew lines.

Step 1: Understanding Skew Lines

Two lines are called skew lines if they do not intersect and are not parallel. In this problem, the lines are given parametrically, allowing us to identify points on each line and the direction vectors of the lines.

Step 2: Understanding the Cross Product

The vector \( \mathbf{n} = \mathbf{n}_1 \times \mathbf{n}_2 \) is the cross product of the direction vectors \( \mathbf{n}_1 \) and \( \mathbf{n}_2 \) of the skew lines. It is perpendicular to both direction vectors.

Step 3: Constructing Vector \( \overrightarrow{PQ} \)

Construct a vector \( \overrightarrow{PQ} \) between points \( P \) and \( Q \) on the two lines. This vector will be used to calculate the projection onto \( \mathbf{n} \).

Step 4: Distance Formula Derivation

The distance \( d \) between the skew lines is the magnitude of the projection of \( \overrightarrow{PQ} \) onto \( \mathbf{n} \). This is calculated using the formula \( d = \frac{|\overrightarrow{PQ} \cdot \mathbf{n}|}{\|\mathbf{n}\|} \).

Step 5: Solving Part (a) – Finding Direction Vectors and a Point on Each Line

For the line equations \( \frac{x-3}{1}=\frac{y+2}{1}=\frac{z-1}{2} \) and \( \frac{x+4}{3}=\frac{y+5}{4}=\frac{z}{5} \), we identify the direction vectors \( \mathbf{n}_1 = (1, 1, 2) \) and \( \mathbf{n}_2 = (3, 4, 5) \), and select \( P = (3, -2, 1) \) and \( Q = (-4, -5, 0) \) as points on respective lines.

Step 6: Part (a) – Cross Product and Dot Product Calculation

Compute the cross product \( \mathbf{n} = \mathbf{n}_1 \times \mathbf{n}_2 \) which results in \( (-3, 1, -1) \). Then, find \( \overrightarrow{PQ} = Q - P \). The dot product \( \overrightarrow{PQ} \cdot \mathbf{n} \) gives the projection on \( \mathbf{n} \).

Step 7: Part (a) – Calculating Distance

Using the derived formula, calculate \( d \) as \( \frac{|\overrightarrow{PQ} \cdot \mathbf{n}|}{\|\mathbf{n}\|} \). Here, \( \|\mathbf{n}\| \) is the magnitude of the vector \( \mathbf{n} \).

Step 8: Solving Part (b) – Identify Direction Vectors and Points

Lines are parameterized as \( x=1+2t, y=-2+3t, z=-4t \) and \( x=3t, y=1+t, z=-5t \), indicating direction vectors \( \mathbf{n}_1 = (2, 3, -4) \) and \( \mathbf{n}_2 = (3, 1, -5) \). Points are selected as \( P = (1, -2, 0) \) and \( Q = (0, 1, 0) \).

Step 9: Part (b) – Cross Product, Dot Product, and Distance

Compute the cross product \( \mathbf{n} = \mathbf{n}_1 \times \mathbf{n}_2 \), find \( \|\mathbf{n}\| \), and calculate \( \overrightarrow{PQ} \cdot \mathbf{n} \). Finally, use the formula \( d = \frac{|\overrightarrow{PQ} \cdot \mathbf{n}|}{\|\mathbf{n}\|} \) to find the distance.

Key Concepts

Cross Product

The cross product is a mathematical operation performed on two vectors in three-dimensional space. The result of a cross product is a third vector, which is both perpendicular to the original two vectors. This is especially useful when determining directions orthogonal to planes. If you have two direction vectors, say \( \mathbf{n}_1 \) and \( \mathbf{n}_2 \), their cross product \( \mathbf{n} = \mathbf{n}_1 \times \mathbf{n}_2 \) gives you a vector orthogonal to both. In the context of skew lines, the cross product helps in finding the shortest distance between them. By calculating the magnitude of this vector, you can work out part of the necessary measures to find out how far apart the two lines are.

Parametric Equations

Parametric equations are a way to express the coordinates of the points making up a geometric object, such as a line, as functions of a parameter. Lines can be described using these parameters, which simplify the representation and calculations of points on the line. For instance, if you have a line defined as \( x = 1 + 2t, y = -2 + 3t, z = -4t \), each variable is expressed in terms of \( t \). As \( t \) changes, the equation maps out all the points along the line. This is especially useful for determining direction vectors of skew lines, aiding in the calculation for the distance between them.

Distance Formula

The distance formula for skew lines provides a way to find the shortest distance between two lines that do not intersect and are not parallel. Using the vector \( \overrightarrow{PQ} \) between points \( P \) and \( Q \) on each line, and the cross product \( \mathbf{n} \) from the lines' direction vectors, the formula is expressed as \( d = \frac{ |\overrightarrow{PQ} \cdot \mathbf{n}| }{ \|\mathbf{n}\| } \). Here, the dot product \( \overrightarrow{PQ} \cdot \mathbf{n} \) measures the projection of \( \overrightarrow{PQ} \) onto \( \mathbf{n} \), and \( \|\mathbf{n}\| \) is the magnitude of the cross product vector. This simple yet powerful formula allows for calculating the shortest path, or distance, between two skew lines.

Dot Product

The dot product is an operation that takes two equal-length sequences of numbers (usually vectors) and returns a single number. This operation is crucial for finding angles between vectors or projecting one vector onto another. Symbolically, for two vectors \( \mathbf{a} = (a_1, a_2, a_3) \) and \( \mathbf{b} = (b_1, b_2, b_3) \), the dot product \( \mathbf{a} \cdot \mathbf{b} \) is calculated as \( a_1b_1 + a_2b_2 + a_3b_3 \). In the problem of finding the distance between skew lines, the dot product of \( \overrightarrow{PQ} \) and the cross product \( \mathbf{n} \) computes a scalar that represents how much \( \overrightarrow{PQ} \) aligns with \( \mathbf{n} \), which is fundamental in deriving the distance formula for skew lines.