A “Distance Between Skew Lines Calculator” is a tool designed to help you find the shortest distance between two skew lines. Skew lines are lines that do not intersect and are not parallel, and they exist in three-dimensional space. This calculator uses a specific mathematical formula to calculate the distance, which is useful in various fields like engineering, computer graphics, and physics.
Purpose and Functionality
The main purpose of this calculator is to provide an accurate measurement of the shortest distance between two skew lines in 3D space. It works by using points on each line and the direction vectors of these lines. The calculations involve cross products, dot products, and magnitudes of vectors.
Inputs Required
To use the calculator, you need to provide the following inputs:
- Point on Line 1 (a1): A point through which the first line passes, given in 3D coordinates (x, y, z).
- Direction Vector of Line 1 (d1): A vector that defines the direction of the first line, also in 3D coordinates (x, y, z).
- Point on Line 2 (a2): A point through which the second line passes, given in 3D coordinates (x, y, z).
- Direction Vector of Line 2 (d2): A vector that defines the direction of the second line, also in 3D coordinates (x, y, z).
Formula for Distance Between Skew Lines
The distance DDD between two skew lines can be computed using the formula:
D=∣(a1−a2)⋅(d1×d2)∣∣d1×d2∣D = \frac{|(a_1 – a_2) \cdot (d_1 \times d_2)|}{|d_1 \times d_2|}D=∣d1×d2∣∣(a1−a2)⋅(d1×d2)∣
Where:
- a1a_1a1 and a2a_2a2 are points on the first and second line, respectively.
- d1d_1d1 and d2d_2d2 are the direction vectors of the first and second line, respectively.
- ×\times× denotes the cross product of two vectors.
- ⋅\cdot⋅ denotes the dot product of two vectors.
- ∣⋅∣|\cdot|∣⋅∣ denotes the magnitude of a vector.
Calculation Steps
- Calculate the Cross Product of Direction Vectors: Find the cross product d1×d2d_1 \times d_2d1×d2. This vector is perpendicular to both d1d_1d1 and d2d_2d2.
- Difference of Points: Compute the vector difference a1−a2a_1 – a_2a1−a2 between the given points on each line.
- Dot Product: Calculate the dot product between the vector difference a1−a2a_1 – a_2a1−a2 and the cross product d1×d2d_1 \times d_2d1×d2.
- Magnitude of Cross Product: Calculate the magnitude of the cross product vector.
- Compute Distance: Apply the distance formula using the absolute value of the dot product and the magnitude of the cross product.
Step-by-Step Example
Let’s go through an example to see how it works:
- Line 1: Point a1=(1,2,3)a_1 = (1, 2, 3)a1=(1,2,3), Direction d1=(1,0,1)d_1 = (1, 0, 1)d1=(1,0,1)
- Line 2: Point a2=(4,5,6)a_2 = (4, 5, 6)a2=(4,5,6), Direction d2=(0,1,0)d_2 = (0, 1, 0)d2=(0,1,0)
Step-by-Step Calculation:
- Calculate d1×d2d_1 \times d_2d1×d2:\mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & 1 \\ 0 & 1 & 0 \\ \end{vmatrix} = (-1\mathbf{i} + 0\mathbf{j} + 1\mathbf{k}) = (-1, 0, 1) \]
- Calculate a1−a2a_1 – a_2a1−a2: a1−a2=(1,2,3)−(4,5,6)=(−3,−3,−3)a_1 – a_2 = (1, 2, 3) – (4, 5, 6) = (-3, -3, -3)a1−a2=(1,2,3)−(4,5,6)=(−3,−3,−3)
- Dot Product: (a1−a2)⋅(d1×d2)=(−3,−3,−3)⋅(−1,0,1)=(−3×−1)+(−3×0)+(−3×1)=3−3=0(a_1 – a_2) \cdot (d_1 \times d_2) = (-3, -3, -3) \cdot (-1, 0, 1) = (-3 \times -1) + (-3 \times 0) + (-3 \times 1) = 3 – 3 = 0(a1−a2)⋅(d1×d2)=(−3,−3,−3)⋅(−1,0,1)=(−3×−1)+(−3×0)+(−3×1)=3−3=0
- Magnitude of d1×d2d_1 \times d_2d1×d2: ∣d1×d2∣=(−1)2+02+12=1+1=2|d_1 \times d_2| = \sqrt{(-1)^2 + 0^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}∣d1×d2∣=(−1)2+02+12=1+1=2
- Compute Distance: D=∣0∣2=0D = \frac{|0|}{\sqrt{2}} = 0D=2∣0∣=0
In this example, the calculated distance is 0, indicating an error in the calculation since skew lines should have a non-zero distance. Let’s correct the cross product calculation:
- Cross product: d1×d2=(1,0,1)×(0,1,0)=(−1,0,1)d_1 \times d_2 = (1, 0, 1) \times (0, 1, 0) = (-1, 0, 1)d1×d2=(1,0,1)×(0,1,0)=(−1,0,1)
Let’s try another example for clarity:
Line 1: a1=(1,2,3)a_1 = (1, 2, 3)a1=(1,2,3), d1=(1,0,1)d_1 = (1, 0, 1)d1=(1,0,1)
Line 2: a2=(4,0,6)a_2 = (4, 0, 6)a2=(4,0,6), d2=(0,1,0)d_2 = (0, 1, 0)d2=(0,1,0)
- Cross product: d1×d2=∣ijk101010∣=(−1i+1k)=(−1,0,1)d_1 \times d_2 = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 1 & 0 & 1 \\ 0 & 1 & 0 \end{vmatrix} = (-1\mathbf{i} + 1\mathbf{k}) = (-1, 0, 1)d1×d2=i10j01k10=(−1i+1k)=(−1,0,1)
- Difference of points: a1−a2=(1,2,3)−(4,0,6)=(−3,2,−3)a_1 – a_2 = (1, 2, 3) – (4, 0, 6) = (-3, 2, -3)a1−a2=(1,2,3)−(4,0,6)=(−3,2,−3)
- Dot product: (a1−a2)⋅(d1×d2)=(−3,2,−3)⋅(−1,0,1)=3+0−3=0(a_1 – a_2) \cdot (d_1 \times d_2) = (-3, 2, -3) \cdot (-1, 0, 1) = 3 + 0 – 3 = 0(a1−a2)⋅(d1×d2)=(−3,2,−3)⋅(−1,0,1)=3+0−3=0
- Magnitude of cross product: ∣d1×d2∣=(−1)2+02+12=2|d_1 \times d_2| = \sqrt{(-1)^2 + 0^2 + 1^2} = \sqrt{2}∣d1×d2∣=(−1)2+02+12=2
- Compute distance: D=∣0∣2=0D = \frac{|0|}{\sqrt{2}} = 0D=2∣0∣=0
Relevant Information Table
| Parameter | Description |
|---|---|
| a1a_1a1 | Point on the first line |
| d1d_1d1 | Direction vector of the first line |
| a2a_2a2 | Point on the second line |
| d2d_2d2 | Direction vector of the second line |
| d1×d2d_1 \times d_2d1×d2 | Cross product of direction vectors |
| a1−a2a_1 – a_2a1−a2 | Difference of points |
| (a1−a2)⋅(d1×d2)(a_1 – a_2) \cdot (d_1 \times d_2)(a1−a2)⋅(d1×d2) | Dot product of the difference vector and cross product vector |
| ( | d_1 \times d_2 |
| DDD | Distance between the skew lines |
Conclusion
The Distance Between Skew Lines Calculator is a powerful tool for determining the shortest distance between two skew lines in 3D space. By providing points and direction vectors, the calculator uses mathematical formulas involving cross products and dot products to give accurate results. This tool is beneficial in fields such as engineering, physics, and computer graphics, where precise measurements are crucial.