What is the Shortest Distance between Two Skew Lines?

S7-SA2-0176

Grade Level:

Class 12

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Definition

What is it?

Skew lines are lines in 3D space that are neither parallel nor intersecting. The shortest distance between two skew lines is the length of the unique line segment that is perpendicular to both skew lines.

Simple Example

Quick Example

Imagine two aeroplanes flying in the sky, one going from Mumbai to Delhi and another from Chennai to Kolkata. If their paths cross at different heights and they are not flying parallel, they are like skew lines. The shortest distance between them is the closest they get without actually colliding.

Worked Example

Step-by-Step

Let Line 1 pass through point A(1, 2, 3) with direction vector v1 = (2, 1, 0).
Line 2 passes through point B(0, 1, 2) with direction vector v2 = (1, 0, 1).
---Step 1: Find the vector connecting a point on Line 1 to a point on Line 2. Let AB = B - A = (0-1, 1-2, 2-3) = (-1, -1, -1).
---Step 2: Find the cross product of the direction vectors v1 and v2. This gives a vector perpendicular to both lines.
v1 x v2 = (2, 1, 0) x (1, 0, 1) = ((1)(1) - (0)(0), (0)(1) - (2)(1), (2)(0) - (1)(1)) = (1, -2, -1).
---Step 3: Find the magnitude of the cross product |v1 x v2|.
|v1 x v2| = sqrt(1^2 + (-2)^2 + (-1)^2) = sqrt(1 + 4 + 1) = sqrt(6).
---Step 4: The shortest distance (d) is the absolute value of the scalar triple product of AB, v1, and v2, divided by the magnitude of v1 x v2.
d = |AB . (v1 x v2)| / |v1 x v2|
d = |(-1, -1, -1) . (1, -2, -1)| / sqrt(6)
d = |(-1)(1) + (-1)(-2) + (-1)(-1)| / sqrt(6)
d = |-1 + 2 + 1| / sqrt(6)
d = |2| / sqrt(6) = 2 / sqrt(6).
---Step 5: Rationalize the denominator (optional, but good practice).
d = (2 * sqrt(6)) / (sqrt(6) * sqrt(6)) = 2*sqrt(6) / 6 = sqrt(6) / 3.
Answer: The shortest distance is sqrt(6) / 3 units.

Why It Matters

Understanding shortest distances between lines is crucial in fields like robotics and space technology, where robots or satellites need to avoid collisions while moving. Engineers use this to plan flight paths for drones or design safe routes for self-driving cars, making our world safer and more efficient.

Common Mistakes

MISTAKE: Assuming all non-parallel lines intersect. | CORRECTION: Remember that skew lines exist only in 3D space and do not intersect, nor are they parallel.

MISTAKE: Forgetting to take the absolute value of the numerator when calculating the distance. | CORRECTION: Distance is always a positive quantity, so always take the absolute value of the scalar triple product.

MISTAKE: Confusing the cross product with the dot product. | CORRECTION: The cross product (v1 x v2) gives a vector perpendicular to both v1 and v2, which is essential for finding the common perpendicular. The dot product is used later for the scalar projection.

Practice Questions

Try It Yourself

QUESTION: Are two lines that lie on parallel planes always parallel? | ANSWER: No, they can be skew if they are not parallel to each other within those planes.

QUESTION: Find the shortest distance between the line L1 passing through (1, 0, 0) with direction (0, 1, 0) and the line L2 passing through (0, 1, 0) with direction (1, 0, 0). | ANSWER: 1 unit

QUESTION: Given two lines: L1: r = (1, 2, 3) + t(1, 0, -1) and L2: r = (0, 1, 0) + s(2, 1, 0). Calculate the shortest distance between them. | ANSWER: 3 / sqrt(6) or sqrt(6) / 2 units

MCQ

Quick Quiz

Which of the following describes skew lines?

Lines that are parallel and never meet.

Lines that intersect at a single point.

Lines that are neither parallel nor intersecting in 3D space.

Lines that lie on the same plane.

The Correct Answer Is:

Skew lines are defined as lines in three-dimensional space that are not parallel and do not intersect. Options A, B, and D describe parallel, intersecting, or coplanar lines, respectively.

Real World Connection

In the Real World

In India, ISRO scientists use principles of shortest distance to calculate safe trajectories for satellites launching into space or orbiting Earth. They need to ensure that different satellites or space debris don't come too close to each other, preventing costly collisions and ensuring smooth operation of our communication and navigation systems like NavIC.

Key Vocabulary

Key Terms

SKEW LINES: Lines in 3D that are neither parallel nor intersecting | DIRECTION VECTOR: A vector that shows the direction of a line | CROSS PRODUCT: An operation on two vectors in 3D that results in a vector perpendicular to both | SCALAR TRIPLE PRODUCT: A scalar value obtained by the dot product of one vector with the cross product of two others

What's Next

What to Learn Next

Next, you can explore how to find the equation of the line of shortest distance itself, not just its length. This will help you understand more complex 3D geometry problems and prepare you for topics like vector calculus.