What is the Shortest Distance Between Two Lines?

S6-SA1-0419

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The shortest distance between two lines is the length of the line segment that is perpendicular to both lines. Imagine two straight roads that don't meet; the shortest way to cross from one road to the other is always at a 90-degree angle.

Simple Example

Quick Example

Imagine two parallel railway tracks in your city. If you want to measure the shortest distance between them, you wouldn't measure diagonally. You would measure straight across, making sure your measuring tape is exactly perpendicular to both tracks.

Worked Example

Step-by-Step

Let's find the shortest distance between two parallel lines: Line 1: y = 2x + 3 and Line 2: y = 2x - 2.
1. Pick a point on Line 1. Let's choose x = 0. Then y = 2(0) + 3 = 3. So, Point P1 is (0, 3).
---2. The shortest distance between two parallel lines is the perpendicular distance from any point on one line to the other line. The formula for the perpendicular distance from a point (x1, y1) to a line Ax + By + C = 0 is |Ax1 + By1 + C| / sqrt(A^2 + B^2).
---3. Rewrite Line 2 (y = 2x - 2) in the form Ax + By + C = 0. This becomes 2x - y - 2 = 0. So, A = 2, B = -1, C = -2.
---4. Now, substitute the point P1 (0, 3) and the values A, B, C into the formula: |(2)(0) + (-1)(3) + (-2)| / sqrt(2^2 + (-1)^2).
---5. Calculate the numerator: |0 - 3 - 2| = |-5| = 5.
---6. Calculate the denominator: sqrt(4 + 1) = sqrt(5).
---7. The shortest distance is 5 / sqrt(5).
---8. Rationalize the denominator: (5 / sqrt(5)) * (sqrt(5) / sqrt(5)) = 5*sqrt(5) / 5 = sqrt(5).
Answer: The shortest distance is sqrt(5) units.

Why It Matters

This concept is crucial in fields like robotics and AI, where robots need to navigate efficiently without colliding. In space technology, ISRO engineers use this to calculate safe distances between satellites or spacecraft trajectories. It helps in planning optimal routes and avoiding obstacles in many engineering designs.

Common Mistakes

MISTAKE: Assuming any distance between two lines is the 'shortest' | CORRECTION: The shortest distance is *always* measured along a line segment that is perpendicular (at 90 degrees) to both original lines.

MISTAKE: Confusing parallel lines with intersecting lines when finding shortest distance | CORRECTION: For intersecting lines, the shortest distance between them is zero because they meet at a point. This concept applies meaningfully to parallel or skew lines.

MISTAKE: Forgetting to convert line equations to the standard form (Ax + By + C = 0) before applying the distance formula | CORRECTION: Always rewrite the line equation as Ax + By + C = 0 to correctly identify A, B, and C for the distance formula.

Practice Questions

Try It Yourself

QUESTION: What is the shortest distance between the parallel lines y = 3x + 1 and y = 3x - 4? | ANSWER: 5 / sqrt(10) units

QUESTION: Find the shortest distance between the point (1, 2) and the line 4x + 3y - 5 = 0. | ANSWER: 3 / 5 units

QUESTION: Two roads are represented by the equations 2x - y + 6 = 0 and 2x - y - 4 = 0. An engineer needs to build a pedestrian bridge perpendicular to both roads. What will be the shortest length of this bridge? | ANSWER: 10 / sqrt(5) = 2*sqrt(5) units

MCQ

Quick Quiz

For two lines to have a non-zero shortest distance between them, they must be:

Intersecting

Perpendicular

Parallel or Skew

Coincident

The Correct Answer Is:

If lines are intersecting or coincident, their shortest distance is 0. If they are perpendicular, they are also intersecting. Only parallel or skew lines (lines not parallel but not intersecting in 3D) have a non-zero shortest distance.

Real World Connection

In the Real World

Imagine a drone delivering a package in a crowded Indian city. It uses this concept to calculate the safest flight path, maintaining a minimum distance from buildings or power lines (which can be thought of as lines in space). Similarly, in making maps for navigation apps like Google Maps or Ola/Uber, engineers use these calculations to find the most efficient routes, avoiding obstacles.

Key Vocabulary

Key Terms

PERPENDICULAR: At a 90-degree angle to another line or surface | PARALLEL LINES: Lines that are always the same distance apart and never meet | SKEW LINES: Lines in 3D space that are not parallel and do not intersect | DISTANCE FORMULA: A mathematical formula used to calculate the length between two points or a point and a line

What's Next

What to Learn Next

Now that you understand distances between lines, explore 'Distance Between a Point and a Plane' or 'Distance Between Two Skew Lines in 3D'. These concepts build on your understanding and are vital for advanced geometry and real-world applications.