Find the shortest distance between the lines r = ( i + 2 j + k ) … - Vidyadip

Question

Find the shortest distance between the lines

$\vec r = \left( {\hat i + 2\hat j + \hat k} \right) + \lambda \left( {\hat i - \hat j + \hat k} \right)$

$\vec r = \left( {2\hat i - \hat j - \hat k} \right) + \mu \left( {2\hat i + \hat j + 2\hat k} \right)$

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Answer

${\vec a_1} = \hat i + 2\hat j + \hat k,{\vec b_1} = \hat i - \hat j + \hat k$

${\vec a_2} = 2\hat i - \hat j - \hat k,{\vec b_2} = 2\hat i + \hat j + 2\hat k$

$d = \left| {\frac{{\left( {{{\vec a}_2} - {{\vec a}_1}} \right).\left( {{{\vec b}_1} \times {{\vec b}_2}} \right)}}{{\left| {{{\vec b}_1} \times {{\vec b}_2}} \right|}}} \right|$

${\vec a_2} - \vec a{_1} = \hat i - 3\hat j - 2\hat k$

${\vec b_1} \times {\vec b_2} = \left| {\begin{array}{*{20}{c}} {\hat i}&{\hat j}&{\hat k} \\ 1&{ - 1}&1 \\ 2&1&2 \end{array}} \right|\\ $

$=\hat i(-2-1)-\hat j(2-2)+\hat k(1+2)$

$ = - 3\hat i + 3\hat k$

$d = \left| {\frac{{\left( {\hat i - 3\hat j - 2\hat k} \right).\left( { - 3\hat i + 3\hat k} \right)}}{{\left| { - 3\hat i + 3\hat k} \right|}}} \right|$

$ = \left| {\frac{{ - 3 - 6}}{{\sqrt {9 + 9} }}} \right| = \left| {\frac{9}{{3\sqrt 2 }}} \right| = \frac{3}{{\sqrt 2 }}$

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