How to Calculate the Shortest Distance Between Two Lines in 2D and 3D?
The concept of distance between two lines is essential in mathematics and helps in solving real-world and exam-level geometry and analytical problems efficiently. Understanding this concept makes coordinate geometry and vectors much easier for students, especially when preparing for board exams and entrance tests.
Understanding Distance Between Two Lines
A distance between two lines refers to the shortest length or separation between two straight lines in a plane or in space. This distance is especially important when the lines are parallel, as the gap between them remains constant everywhere, and also when lines are skew (not parallel and non-intersecting) in 3D geometry. You will commonly find distance between two lines used in analytical geometry, coordinate geometry, and vector mathematics. It helps calculate the minimum gap, perpendicular projections, and solve many syllabus-based questions from class 11 and 12 as well as competitive exams.
Formula Used in Distance Between Two Lines
The standard formula for the distance between two parallel lines in 2D (straight lines in the form Ax + By + C = 0) is:
\( d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \)
For lines in the form y = mx + c, the formula becomes:
\( d = \frac{|c_1 - c_2|}{\sqrt{1 + m^2}} \)
In 3D, for skew lines (non-parallel, non-intersecting), the distance between two lines vector formula is:
\( d = \frac{|(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|} \),
where \( \vec{a_1}, \vec{a_2} \) are position vectors on the two lines, and \( \vec{b_1}, \vec{b_2} \) are their direction vectors. This is crucial for the shortest distance between two lines in 3D.
Here’s a helpful table summarising key formulas for different cases of distance between two lines:
Distance Between Two Lines Table
| Type of Lines | Standard Form | Distance Formula |
|---|---|---|
| Parallel Lines (2D) | Ax + By + C1 = 0 Ax + By + C2 = 0 |
\( \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \) |
| Parallel Lines (Slope-Intercept) | y = mx + c1, y = mx + c2 | \( \frac{|c_1 - c_2|}{\sqrt{1 + m^2}} \) |
| Skew/Parallel (3D, Vector) | Vector form | \( \frac{|(\vec{a_2} - \vec{a_1}) \cdot (\vec{b_1} \times \vec{b_2})|}{|\vec{b_1} \times \vec{b_2}|} \) |
This table shows how the pattern of distance between two lines can be solved in various scenarios using the appropriate formula.
Worked Example – Solving a Distance Between Parallel Lines Problem
Let’s solve a problem step by step to understand this concept clearly:
1. The equations of two lines are: 3x + 4y = 9 and 3x + 4y = 15/2
2. First, check if lines are parallel: Compare the coefficients. Both have A = 3 and B = 4, so the lines are parallel.
3. Standard form:
Line 1: 3x + 4y - 9 = 0 (C1 = -9)
Line 2: 3x + 4y - 15/2 = 0 (C2 = -15/2)
4. Use the formula:
\( d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}} \)
Substitute values:
\( d = \frac{|-9 - (-15/2)|}{\sqrt{3^2 + 4^2}} \)
\( d = \frac{|-9 + 7.5|}{5} \)
\( d = \frac{1.5}{5} \)
5. Final Answer:
The distance between the lines is 0.3 units.
Practice Problems
- Find the distance between the lines 2x - 3y + 7 = 0 and 2x - 3y - 11 = 0.
- Calculate the shortest distance between two lines with equations x + y + 4 = 0 and x + y - 2 = 0.
- In 3D, find the distance between skew lines: \( \vec{r_1} = \vec{a_1} + t\vec{b_1} \), \( \vec{r_2} = \vec{a_2} + s\vec{b_2} \), where
\( \vec{a_1} = (1,0,0), \, \vec{b_1} = (1,1,0) \),
\( \vec{a_2} = (0,1,1), \, \vec{b_2} = (0,1,1) \). - If the distance between lines 5x - 12y + 2 = 0 and 5x - 12y + k = 0 is 5/13, find the value of k.
Common Mistakes to Avoid
- Confusing the formula for distance between parallel lines with the distance from a point to a line.
- Not simplifying equations to standard form before applying the formula.
- Switching values of C1 and C2 (always use absolute value in numerator).
- Attempting to use parallel line formulas for non-parallel or intersecting lines.
Real-World Applications
The concept of distance between two lines is useful in urban planning (for shortest road distance), navigation (finding the minimal gap between paths), physics (distance between wires, rails), and even computer graphics. Vedantu helps students visualise and solve such geometric questions, making preparation easier for practical exams.
Page Summary
We explored the idea of distance between two lines, including 2D and 3D cases, formula applications, and worked examples. Practice more such problems on Vedantu and use these formulas for exam confidence and practical problem-solving skills.
Related Topics to Explore
- Distance Between Two Points
- Straight Lines
- Equation of a Line
- Parallel Lines
- Distance Between Two Parallel Lines
- Angle Between Two Lines
- Properties of Parallel Lines
- Coordinate Geometry
- Line Segment
- 3D Formulas
- Perpendicular Distance of a Point from a Line
FAQs on Distance Between Two Lines: Concepts & Formulas Explained
1. What is the formula for distance between two lines?
The distance between two lines is the shortest perpendicular gap between them. For parallel lines in 2D, the formula is: Distance = \frac{|C_2 - C_1|}{\sqrt{A^2 + B^2}},
where the lines are in the form Ax + By + C = 0. For skew lines in 3D, vector formulas involving direction vectors and cross products are used to find the minimum distance.
2. How to calculate the distance between two parallel lines?
To calculate the distance between two parallel lines, ensure their slopes are equal. Using their standard form Ax + By + C = 0, the distance is computed as:d = \frac{|C_1 - C_2|}{\sqrt{A^2 + B^2}}.
This gives the length of the perpendicular segment connecting the two lines.
3. How do you find the distance between two lines in 3D?
In 3D coordinate geometry, the distance between two lines that are not parallel or intersecting (i.e., skew lines) is calculated using vector algebra. The formula involves the cross product of their direction vectors \vec{d_1} and \vec{d_2} and a vector \vec{c} connecting points on each line:d = \frac{|(\vec{c} \cdot (\vec{d_1} \times \vec{d_2}))|}{|\vec{d_1} \times \vec{d_2}|}.
This gives the shortest distance between the skew lines.
4. What is distance between two lines in vector form?
The vector form represents lines as \vec{r} = \vec{a} + \lambda \vec{b}, where \vec{a} is a point on the line and \vec{b} is its direction vector. The distance between two such lines is found by:
1. Finding a vector \vec{c} connecting points on both lines,
2. Taking the magnitude of the scalar triple product of \vec{c} with the cross product of direction vectors,
3. Dividing by the magnitude of the cross product.
This method is essential in skew line distance calculations.
5. Is there a calculator for distance between two lines?
Yes, there are online calculators available to quickly find the distance between two lines in both 2D and 3D. These calculators require you to input line equations or points and direction vectors and provide step-by-step solutions. Using such tools can aid in fast and accurate exam revision and practice.
6. Why do students confuse 2D and 3D distance formulas?
Students often confuse 2D and 3D distance formulas because the concepts seem similar but differ in application:
- 2D distance uses simple perpendicular distance formulas between parallel lines.
- 3D distance requires vector operations like cross products to handle skew lines.
Understanding the geometric differences and practicing both cases can clarify these confusions.
7. What mistakes occur when lines are not parallel?
When lines are not parallel, students often mistakenly apply the parallel line distance formula, leading to incorrect results. In such cases:
- If lines intersect, the distance is zero.
- For skew lines in 3D, one must use vector cross product methods.
Failing to identify the line relationships causes common errors in distance calculations.
8. Why is vector approach important in JEE/advanced exams?
The vector approach is crucial for JEE and other advanced exams as it provides:
- A systematic way to handle distances in 3D geometry.
- Tools to tackle skew lines and their shortest distances.
- Connections to other concepts like direction ratios and scalar projections.
Mastering vectors enhances problem-solving ability in analytical geometry problems.
9. When is the distance zero between two lines?
The distance between two lines is zero when:
- The lines intersect at a point, i.e., they share a common point.
- They are coincident, meaning they lie exactly on top of each other.
In both situations, there is no gap separating the lines.
10. How can visual diagrams help in understanding the concept?
Visual diagrams support understanding by:
- Showing the geometric relationship between lines (parallel, intersecting, skew).
- Illustrating perpendicular distances and projections.
- Aiding memory retention through visual learning.
- Increasing clarity especially on mobile devices with clear, scaled sketches.
Incorporating visuals makes grasping formulas and concepts easier for students.
