Formula For Distance Between 2 Skew Lines

Distance Between Two Skew Lines 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 Distance Between Two Skew Lines?

Skew lines are lines in three-dimensional space that are neither parallel nor intersecting. The distance between two skew lines is the length of the shortest line segment connecting them, which is perpendicular to both lines.

2. How Does The Calculator Work?

The calculator uses the vector formula for distance between skew lines:

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

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 calculates the perpendicular distance between two skew lines using vector operations. The cross product gives a vector perpendicular to both lines, and the dot product projects the vector between points onto this perpendicular direction.

3. Mathematical Derivation

Details: The distance is found by taking the absolute value of the scalar projection of the vector connecting points on the two lines onto the unit vector perpendicular to both lines. This perpendicular vector is obtained from the cross product of the direction vectors.

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-zero and lines are not parallel (cross product should not be zero).

5. Frequently Asked Questions (FAQ)

Q1: What if the lines are parallel?
A: If lines are parallel, the cross product is zero and the formula becomes undefined. For parallel lines, use the distance formula between a point and a line.

Q2: What if the lines intersect?
A: If lines intersect, the distance is zero since they share a common point.

Q3: Can this formula be used in 2D?
A: In 2D, lines are either parallel or intersecting, so the concept of skew lines doesn't apply. All 2D lines are coplanar.

Q4: What are practical applications of this calculation?
A: Used in computer graphics, robotics, mechanical engineering, and physics for collision detection, path planning, and spatial analysis.

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