Skew Lines Distance Formula

Skew Lines Distance Formula:

\[ d = \frac{|(\vec{P_2} - \vec{P_1}) \cdot (\vec{d_1} \times \vec{d_2})|}{|\vec{d_1} \times \vec{d_2}|} \]

Point P1 (x, y, z):

Direction Vector d1 (x, y, z):

Point P2 (x, y, z):

Direction Vector d2 (x, y, z):

Distance Between Skew Lines:

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1. What is the Skew Lines Distance Formula?

The Skew Lines Distance Formula calculates the shortest distance between two non-parallel, non-intersecting lines in three-dimensional space. Skew lines are lines that do not lie in the same plane and never meet.

2. How Does the Calculator Work?

The calculator uses the skew lines distance formula:

\[ d = \frac{|(\vec{P_2} - \vec{P_1}) \cdot (\vec{d_1} \times \vec{d_2})|}{|\vec{d_1} \times \vec{d_2}|} \]

Where:

\( \vec{P_1} \) — Point on first line
\( \vec{P_2} \) — Point on second line
\( \vec{d_1} \) — Direction vector of first line
\( \vec{d_2} \) — Direction vector of second line
\( \times \) — Cross product
\( \cdot \) — Dot product
\( |\vec{v}| \) — Magnitude of vector

Explanation: The formula finds the distance as the absolute value of the scalar triple product divided by the magnitude of the cross product of the direction vectors.

3. Importance of Skew Lines Distance

Details: Calculating distance between skew lines is essential in 3D geometry, computer graphics, robotics, engineering design, and spatial analysis where determining clearance between non-intersecting objects is required.

4. Using the Calculator

Tips: Enter coordinates for one point on each line and the direction vectors for both lines. Ensure direction vectors are non-parallel for valid results. All values should be real numbers.

5. Frequently Asked Questions (FAQ)

Q1: What are skew lines?
A: Skew lines are lines in three-dimensional space that are not parallel and do not intersect. They lie in different planes.

Q2: What if the lines are parallel?
A: If lines are parallel, the cross product of direction vectors becomes zero and the formula is undefined. Use the parallel lines distance formula instead.

Q3: Can this formula be used for intersecting lines?
A: For intersecting lines, the distance is zero. The formula will return a very small value or zero depending on input precision.

Q4: What units does the distance have?
A: The distance has the same units as the input coordinates (meters, feet, etc.).

Q5: How accurate is this calculation?
A: The calculation is mathematically exact for the given inputs. Accuracy depends on the precision of your input values.