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.

✍️Mathematical Proof

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🧄Proof and Derivation-2

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