Shortest Distance Between Two Skew Lines

In three-dimensional geometry, skew lines are two lines that do not intersect and are not parallel. Finding the shortest distance between two skew lines is a common problem in vector calculus and has practical applications in fields such as engineering and physics.

Understanding Skew Lines

Before we delve into the method of finding the shortest distance, let's understand what skew lines are:

• Skew Lines: Two lines in three-dimensional space that do not intersect and are not parallel.

Shortest Distance

The shortest distance between two skew lines is the length of the line segment perpendicular to both lines. This segment is often referred to as the common perpendicular.

Mathematical Approach

To find the shortest distance between two skew lines, we can use vector algebra. Let's consider two skew lines given by their vector equations:

$$ \begin{align*} \text{Line } L_1 &: \vec{r} = \vec{a}_1 + \lambda \vec{b}_1 \ \text{Line } L_2 &: \vec{r} = \vec{a}_2 + \mu \vec{b}_2 \end{align*} $$

where $\vec{a}_1$ and $\vec{a}_2$ are position vectors of points on $L_1$ and $L_2$, respectively, and $\vec{b}_1$ and $\vec{b}_2$ are the direction vectors of $L_1$ and $L_2$, respectively. $\lambda$ and $\mu$ are scalar parameters.

Formula for Shortest Distance

The shortest distance $d$ between the two skew lines can be found using the formula:

$$ d = \frac{|\vec{a}_2 - \vec{a}_1 \cdot (\vec{b}_1 \times \vec{b}_2)|}{|\vec{b}_1 \times \vec{b}_2|} $$

where $\times$ denotes the cross product of two vectors.

Steps to Find the Shortest Distance

1. Find Direction Vectors: Determine the direction vectors $\vec{b}_1$ and $\vec{b}_2$ of the lines $L_1$ and $L_2$.
2. Compute the Cross Product: Calculate the cross product $\vec{b}_1 \times \vec{b}_2$ to find a vector perpendicular to both direction vectors.
3. Find the Connecting Vector: Determine the vector connecting a point on $L_1$ to a point on $L_2$, which is $\vec{a}_2 - \vec{a}_1$.
4. Apply the Formula: Use the formula for the shortest distance to calculate $d$.

Table of Differences and Important Points

Aspect	Description for Skew Lines
Intersection	Skew lines do not intersect.
Parallelism	Skew lines are not parallel.
Shortest Distance	Perpendicular to both lines.
Calculation Method	Use cross product and vector algebra.
Formula Applicability	Only for skew lines in 3D space.

Examples

Example 1

Let's consider two skew lines $L_1$ and $L_2$ with the following vector equations:

$$ L_1: \vec{r} = \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} + \lambda \begin{pmatrix} 1 \ -1 \ 1 \end{pmatrix}, \quad L_2: \vec{r} = \begin{pmatrix} 4 \ 5 \ 6 \end{pmatrix} + \mu \begin{pmatrix} 0 \ 1 \ -1 \end{pmatrix} $$

Solution:

1. The direction vectors are $\vec{b}_1 = \begin{pmatrix} 1 \ -1 \ 1 \end{pmatrix}$ and $\vec{b}_2 = \begin{pmatrix} 0 \ 1 \ -1 \end{pmatrix}$.
2. The cross product $\vec{b}_1 \times \vec{b}_2 = \begin{pmatrix} 1 \ -1 \ 1 \end{pmatrix} \times \begin{pmatrix} 0 \ 1 \ -1 \end{pmatrix} = \begin{pmatrix} -2 \ -1 \ 1 \end{pmatrix}$.
3. The connecting vector $\vec{a}_2 - \vec{a}_1 = \begin{pmatrix} 4 \ 5 \ 6 \end{pmatrix} - \begin{pmatrix} 1 \ 2 \ 3 \end{pmatrix} = \begin{pmatrix} 3 \ 3 \ 3 \end{pmatrix}$.
4. The shortest distance $d = \frac{|\begin{pmatrix} 3 \ 3 \ 3 \end{pmatrix} \cdot \begin{pmatrix} -2 \ -1 \ 1 \end{pmatrix}|}{|\begin{pmatrix} -2 \ -1 \ 1 \end{pmatrix}|} = \frac{|-6 -3 +3|}{\sqrt{(-2)^2 + (-1)^2 + 1^2}} = \frac{6}{\sqrt{6}} = \sqrt{6}$.

Therefore, the shortest distance between the two skew lines is $\sqrt{6}$ units.

Example 2

For a more conceptual example, imagine two skew lines represented by the edges of a box, where one line is along the top front edge and the other along the bottom back edge. The shortest distance between these lines would be the line segment that connects the closest points on these edges and passes through the interior of the box, perpendicular to both edges.

Conclusion

Finding the shortest distance between two skew lines involves understanding the concept of skew lines, using vector algebra, and applying the formula involving the cross product of direction vectors. This concept is essential in various applications, including engineering design and physics simulations.