Shortest Distance between two Lines in 3D Calculator

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$\bold{Table \space of \space Content}$

• 1. Introduction to the Shortest distance between two Skew lines in 3-D
• 2. What is the Formulae used ?
• 3. How do I calculate the Shortest distance between two Skew lines in 3-D?
• 4. Why choose our Shortest distance between two Skew lines in 3-D Calculator?
• 5. A Video for explaining this concept
• 6. How to use this calculator ?
• 7. Solved Examples
• 8. Frequently Asked Questions (FAQs)
• 9. What are the real-life applications?
• 10. Conclusion

1. Introduction to the Shortest distance between two Skew lines in 3-D

Embarking on a three-dimensional exploration, we explore the intriguing challenge of finding the shortest distance between two lines. Join us as we navigate this spatial puzzle, unveiling the definition, practical applications, and a step-by-step guide to determine the closest connection between parallel paths in the three-dimensional realm.
$\bold{Definition}$
The shortest distance between two lines in three-dimensional space represents the perpendicular distance between any two points on the lines. It encapsulates the concept of the smallest separation between two non-intersecting lines, creating a link that minimizes the spatial gap in three-dimensional coordinates.
Skew lines in 3D space are non-parallel and non-intersecting lines, maintaining a constant separation without converging or diverging.

2. What is the Formulae used?

$\bold{Cartesian \space Method}$
If the line $L_1: \bold{\frac{x - x_1}{a_1} = \frac{y - y_1}{b_1} = \frac{z - z_1}{c_1}}$ & $L_2: \bold{\frac{x - x_2}{a_2} = \frac{y - y_2}{b_2} = \frac{z - z_2}{c_2}}$

then shortest distance between $L_1$ & $L_2$ is given by d = $|\bold{\frac{\begin{vmatrix} x_2 - x_1 & y_2 - y_1 & z_2 - z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \end{vmatrix}}{\sqrt{(a_1.b_2 - a_2.b_1)^2 + (b_1.c_2 - b_2.c_1)^2 + (c_1.a_2 - c_2.a_1)^2}}}|$

The condition required is the lines must be $\bold{non-intersecting}$, and it's essential to have a clear understanding of the direction vectors of both lines. This ensures a well-defined perpendicular distance. $\bold{Vector \space Method}$
If the line $L_1:$ $\vec{r} = \vec{a_1} + \lambda\vec{b_1}$ & $L_2:$ $\vec{r} = \vec{a_2} + \mu\vec{b_2}$
Then shortest distance between $L_1$ & $L_2$ is given by d = $|\bold{\frac{(\vec{b_1} x \vec{b_2}).(\vec{a_2} -\vec{a_1})}{|\vec{b_1}x\vec{b_2}|}}|$

3. How do I calculate the Shortest distance between two Skew lines in 3-D?

Write the equation of the line in either vector form or Cartesian form.
Use the above-given formulas to find the shortest distance between lines in 3-D.

4. Why choose the Shortest distance between two Skew lines in 3-D Calculator?

$\bold{Easy \space \space to \space Use}$
Our calculator page provides a user-friendly interface that makes it accessible to both students and professionals. You can quickly input your square matrix and obtain the matrix of minors within a fraction of a second.

$\bold{Time \space Saving \space By \space automation}$
Our calculator saves you valuable time and effort. You no longer need to manually calculate each cofactor, making complex matrix operations more efficient.

$\bold{Accuracy \space and \space Precision}$
Our calculator ensures accurate results by performing calculations based on established mathematical formulas and algorithms. It eliminates the possibility of human error associated with manual calculations.

$\bold{Versatility}$
Our calculator can handle all input values like integers, fractions, or any real number.

$\bold{Complementary \space Resources}$
Alongside this calculator, our website offers additional calculators related to Pre-algebra, Algebra, Precalculus, Calculus, Coordinate geometry, Linear algebra, Chemistry, Physics, and various algebraic operations. These calculators can further enhance your understanding and proficiency.

5. A video based on the concept of how to find the Shortest distance between Skew two lines in 3-D.

6. How to use this calculator

This calculator will help you to find the Shortest distance between two Skew lines in 3-D.
In the given input boxes, you have to put the value of the equation of lines.
After clicking on the Calculate button, a step-by-step solution will be displayed on the screen.
You can access, download, and share the solution.

7. Solved Examples

$\bold{Question:1}$
Find the Shortest distance between two Skew lines $L_1: \bold{\frac{x + 1}{7} = \frac{y + 1}{-6} = \frac{z + 1}{1}}$ & $L_2: \bold{\frac{x - 3}{1} = \frac{y - 5}{-2} = \frac{z - 7}{1}}$.
$\bold{Solution}$
We will use the Cartesian form formula to find the distance.
Enter the values in the above calculator, and you will get step-by-step solutions.
Shortest distance d = $2\sqrt{29}$

$\bold{Question:2}$
Find the Shortest distance between two Skew lines $L_1:$ $\vec{r} = (i + 2j - 4k) + \lambda(2i + 3j + 6k)$ & $L_2:$ $\vec{r} = (3i + 3j - 5k) + \mu(i + 4j + 2k)$.
$\bold{Solution}$
We will use the vector form formula to find the distance.
Enter the values in the above calculator, and you will get step-by-step solutions.
Shortest distance d = $\frac{39}{\sqrt{353}}$

8. Frequently Asked Questions (FAQs):-

Can intersecting lines have the shortest distance?

No, the lines must be non-intersecting for a meaningful shortest distance.

Do both lines need to be in vector form for this calculation?

Expressing the lines in vector form allows for easy determination of direction vectors, simplifying the calculation.

What if the lines are coincident or parallel?

Coincident lines have a zero distance, while parallel lines maintain a constant separation.

Are direction vectors crucial for the calculation?

Yes, direction vectors provide the orientation of the lines, ensuring a well-defined perpendicular distance.

Can this formula be extended to more than two lines?

The formula is specific to finding the distance between two lines and is not directly applicable to more than two lines.

9. What are the real-life applications?

In computer graphics, determining the shortest distance between two 3D lines is essential for rendering realistic images and ensuring accurate placement of objects in a three-dimensional space.

10. Conclusion

As we conclude our exploration of finding the shortest distance between two lines in 3D, we recognize its significance in understanding spatial relationships. Embrace the formula's simplicity and applications in diverse fields, from computer graphics to robotics. Though rooted in vector calculations, this geometric concept is a crucial tool, providing insights into the closest approach to non-intersecting lines in the vast landscapes of three-dimensional coordinates.

This blog is written by Neetesh Kumar

If you have any suggestions regarding the improvement of the content of this page, please write to me at My Official Email Address: [email protected]

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