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L1 మరియు L2  అనేవి సరళరేఖలు. వీటి సమీకరణాలు వరుసగా \(\rm \vec{r}=\vec{a}_1 +\lambda \vec{b}\) మరియు \(\rm r=\vec{a}_2 + \mu\vec{b}\) . అయిన ఈ రెండు రేఖల మధ్యనుండే అతి తక్కువ దూరం?

  1. \(\rm d=\left|\dfrac{\vec{b}\times (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\)
  2. \(\rm d=\left|\dfrac{\vec{b}\cdot (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\)
  3. \(\rm d=\left|\dfrac{\vec{a}_1\times (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\)
  4. \(\rm d=\left|\dfrac{\vec{a}_2\times (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\)

Answer (Detailed Solution Below)

Option 1 : \(\rm d=\left|\dfrac{\vec{b}\times (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\)
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Detailed Solution

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పద్ధతి:

  • రెండు రేఖలు సమాంతరంగా ఉంటే, వాటి మధ్య దూరం స్థిరంగా ఉంటుంది.
  • రెండు సమాంతర రేఖల మధ్య దూరం\(\rm \vec{r}=\vec{a}_1 +\lambda \vec{b}\) and \(\rm r=\vec{a}_2 + \mu\vec{b}\) : \(\rm d=\left|\dfrac{\vec{b}\times (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\).

 

సాధన:

రెండు సమాంతర రేఖల మధ్య దూరం కోసం సూత్రాన్ని ఉపయోగించిన, \(\rm d=\left|\dfrac{\vec{b}\times (\vec{a}_2-\vec{a}_1)}{|\vec{b}|}\right|\).

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