Finding the Shortest Distance and Proving Orthogonality for Skew Lines

Finding the shortest distance between two skew lines relies on minimizing the magnitude of the difference vector, $d(t, s)$, which connects arbitrary points on both lines. This minimization is achieved using calculus by setting the partial derivatives of the squared magnitude, $|d|^2$, with respect to $t$ and $s$ to zero, resulting in a system of linear equations. Solving this system yields the optimal parameters ( $t=2.5, s=1$ ) that define the points of closest approach and the minimum distance, $\sqrt{1.5}$. Crucially, the mathematical proof confirms that the difference vector corresponding to this shortest distance, $d(2.5,1)$, is necessarily orthogonal (dot product is zero) to the direction vectors of both lines, which serves as the fundamental geometric property governing the shortest connection.

🎬Narrated Video

https://youtu.be/3PVGKwnhExw

🏗️Structural clarification of Poof and Derivation

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🗒️Downloadable Files - Recursive updates

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