Question

Find the shortest distance between the lines:
$straight r with rightwards arrow on top space equals space straight i with hat on top space minus space 7 space straight j with hat on top space minus space 2 space straight k with hat on top space plus space straight lambda space open parentheses stack straight i space with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top close parentheses$
and    $straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 2 space straight k with hat on top space plus space straight mu space open parentheses negative straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top close parentheses$

Solution

The equations of two lines are

straight r with rightwards arrow on top space equals space straight i with hat on top space minus space 7 space straight j with hat on top space minus space 2 space straight k with hat on top space plus space straight lambda space open parentheses stack straight i space with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top close parentheses

...(1)
and

straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 2 space straight k with hat on top space plus space straight mu space open parentheses negative straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top close parentheses

...(2)
Comparing these equations with

straight r with rightwards arrow on top space equals space stack straight a subscript 1 with rightwards arrow on top space plus space straight lambda stack straight b subscript 1 with rightwards arrow on top space space and space straight r with rightwards arrow on top space equals space stack straight a subscript 2 with rightwards arrow on top space plus space straight mu space stack straight b subscript 2 with rightwards arrow on top comma space we space get comma

stack straight a subscript 1 with rightwards arrow on top space equals space straight i with hat on top space minus space 7 space straight j with hat on top space minus space 2 space straight k with hat on top comma space space space space space stack straight b subscript 1 with rightwards arrow on top space equals space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top stack straight a subscript 2 with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space minus space 2 space straight k with hat on top comma space space stack straight b subscript 2 with rightwards arrow on top space equals space minus straight i with hat on top space plus space 2 space straight j with hat on top space plus space straight k with hat on top stack straight a subscript 2 with rightwards arrow on top space minus space stack straight a subscript 1 with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space space 11 space straight j with hat on top

stack straight b subscript 1 with rightwards arrow on top space cross times space stack straight b subscript 2 with rightwards arrow on top space equals space open vertical bar table row cell straight i with hat on top end cell cell straight j with hat on top end cell cell straight k with hat on top end cell row 1 3 2 row cell negative 1 end cell 2 1 end table close vertical bar space equals space left parenthesis negative 3 minus 4 right parenthesis space straight i with hat on top space minus space left parenthesis 1 plus 2 right parenthesis space straight j with hat on top space plus space left parenthesis 2 plus 3 right parenthesis space straight k with hat on top

equals space minus straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top

open vertical bar stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top close vertical bar space equals space square root of 1 plus 9 plus 25 end root space equals space square root of 35

S.D. =

open vertical bar fraction numerator left parenthesis stack straight a subscript 2 with rightwards arrow on top minus stack straight a subscript 1 with rightwards arrow on top right parenthesis. space open parentheses stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top close parentheses over denominator open vertical bar stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top close vertical bar end fraction close vertical bar space equals space open vertical bar fraction numerator left parenthesis 2 space straight i with hat on top space plus space 11 space straight j with hat on top right parenthesis. space left parenthesis negative straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis over denominator square root of 35 end fraction close vertical bar

equals space open vertical bar fraction numerator left parenthesis 2 right parenthesis thin space left parenthesis negative 1 right parenthesis thin space plus space left parenthesis 11 right parenthesis thin space left parenthesis negative 3 right parenthesis space plus space left parenthesis 0 right parenthesis thin space left parenthesis 5 right parenthesis over denominator square root of 35 end fraction close vertical bar space equals space open vertical bar negative fraction numerator 35 over denominator square root of 35 end fraction close vertical bar space equals space square root of 35

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