Finding the Shortest Distance and Proving Orthogonality for Skew Lines

The shortest distance between two skew lines is defined by a vector that is orthogonal (perpendicular) to both lines. The analysis proves this mathematically using calculus and dot products, while the interactive demo provides a visual confirmation, showing that any non-orthogonal vector results in a longer distance. This principle of orthogonality is the core geometric insight for solving such problems.

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✍️Mathematical Proof

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Difference Vector for Shortest Distance: The problem demonstrates that the distance between two lines can be represented by a single difference vector d(t, s). Finding the shortest distance is equivalent to finding the minimum magnitude of this vector by using calculus and partial derivatives.
Calculus to Find a Minimum: The use of partial derivatives $\left(\frac{\partial}{\partial t}\right.$ and $\left.\frac{\partial}{\partial s}\right)$ set to zero is a powerful mathematical technique for finding the parameters ( $t$ and $s$ ) that result in the minimum squared distance. This is a fundamental application of multi-variable calculus to optimization problems.
Orthogonality Condition: The analysis confirms a critical geometric principle: the shortest distance vector between two skew lines is always orthogonal (perpendicular) to the tangent vectors of both lines. This is a key geometric insight that can be used to solve such problems.
Proof via Dot Product: The proof of orthogonality is elegantly shown using the dot product. When the dot product of two vectors equals zero, it proves that the vectors are perpendicular. This provides a direct and verifiable way to confirm the geometric relationship.

🎬Demonstration

The shortest distance between two skew lines is always measured along a line segment that is perpendicular (orthogonal) to both of the original lines. The interactive demo visually proves this. The static green vector, representing the shortest distance, is perpendicular to both the yellow tangent vector (on the blue line) and the orange tangent vector (on the red line) at the points of closest approach. By using the sliders, you can see that any other vector connecting the two lines is not perpendicular to both lines, and its length is always greater than the shortest distance. This reinforces the fundamental principle that the minimum distance is achieved when the connecting vector is orthogonal to the direction vectors of both lines.

Interactive Visualization of Shortest Distance Between Skew Lines