Skew Lines Calc 3

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 Skew Lines Distance?

Skew lines are lines in three-dimensional space that are neither parallel nor intersecting. The distance between 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:

\( \vec{P_1}, \vec{P_2} \) — Points on each line
\( \vec{d_1}, \vec{d_2} \) — Direction vectors of each line
\( \times \) — Cross product
\( \cdot \) — Dot product
\( |\vec{v}| \) — Magnitude of vector

Explanation: The formula calculates the perpendicular distance between two lines in 3D space using vector operations.

3. Mathematical Formula

Details: The numerator represents the volume of the parallelepiped formed by the vectors, while the denominator gives the area of the base parallelogram.

4. Using The Calculator

Tips: Enter coordinates for points P1 and P2 on each line, and direction vectors d1 and d2. All values must be valid real numbers.

5. Frequently Asked Questions (FAQ)

Q1: What are skew lines?
A: Skew lines are non-parallel, non-intersecting lines in three-dimensional space that lie in different planes.

Q2: When does this formula not work?
A: The formula fails when lines are parallel (cross product magnitude is zero) or when they intersect (distance is zero).

Q3: What if the lines are parallel?
A: For parallel lines, use the distance formula between a point and a line, or find the distance between the two parallel planes containing the lines.

Q4: Can this be used for 2D lines?
A: In 2D, lines are either parallel or intersecting, so this formula is specifically for 3D space.

Q5: What units does the distance use?
A: The distance is in the same units as the input coordinates (meters, centimeters, etc.).