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JEE Main 2025 · Mathematics · Three Dimensional Geometry · hard

The shortest distance between the line through the origin with direction ratios 1,1,1 and the line through the point (1,2,0) with direction ratios…

Single correct option

The shortest distance between the line through the origin with direction ratios 1,1,11,1,1 and the line through the point (1,2,0)(1,2,0) with direction ratios 1,−1,01,-1,0 is
  1. A. 32\dfrac{3}{2}
  2. B. 3\sqrt3
  3. C. 62\dfrac{\sqrt6}{2}
  4. D. 16\dfrac{1}{\sqrt6}
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Answer: (C) 62\dfrac{\sqrt6}{2}

Take d⃗1=(1,1,1)\vec d_1=(1,1,1), d⃗2=(1,−1,0)\vec d_2=(1,-1,0). Then d⃗1×d⃗2=(1,1,−2)\vec d_1\times\vec d_2=(1,1,-2) with magnitude 6\sqrt6. The vector joining the points is (1,2,0)(1,2,0), and its dot product with the cross product is 1+2+0=31+2+0=3. Distance =36=62=\dfrac{3}{\sqrt6}=\dfrac{\sqrt6}{2}.
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