Multivariable and Vector Calculus · Geometry, vectors, and vector-valued functions
Locate the shortest connector between two skew lines
Problem
Consider \[ L_1:\ \mathbf p(s)=s(1,1,0), \qquad L_2:\ \mathbf q(t)=(1,2,0)+t(1,0,1). \] Prove that the lines are skew. Find the unique parameters $s,t$ minimizing $\|\mathbf p(s)-\mathbf q(t)\|$, then give the closest points, exact distance, and directed connector from $L_2$ to $L_1$. Verify that the connector is perpendicular to both directions. Explain why solving only one perpendicularity equation cannot certify the minimum.
Hint
Write $\mathbf d(s,t)=\mathbf p(s)-\mathbf q(t)$ before expanding its squared norm.
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