Three Dimensional Geometry

Find the shortest distance between the lines: and - Wired Faculty

Question

Find the shortest distance between the lines:
$straight r with rightwards arrow on top space equals space left parenthesis 8 plus 3 space straight lambda right parenthesis space straight i with hat on top space minus space left parenthesis 9 plus 16 space straight lambda right parenthesis space straight j with hat on top space plus space left parenthesis 10 plus space 7 space straight lambda right parenthesis space straight k with hat on top$
and $straight r with rightwards arrow on top space equals space 15 space straight i with hat on top space plus space 29 space straight j with hat on top space plus space 5 space straight k with hat on top space plus space straight mu space open parentheses 3 space straight i with hat on top space plus space 8 space straight j with hat on top space minus space 5 space straight k with hat on top close parentheses.$

Solution

The equation of the two lines are

straight r with rightwards arrow on top space equals space left parenthesis 8 plus 3 space straight lambda right parenthesis space straight i with hat on top space minus space left parenthesis 9 plus 16 space straight lambda right parenthesis space straight j with hat on top space plus space left parenthesis 10 plus 7 space straight lambda right parenthesis straight k with hat on top

straight r with rightwards arrow on top space equals space 8 space straight i with hat on top space minus space 9 space straight j with hat on top space plus space 10 space straight k with hat on top space plus space straight lambda space open parentheses 3 space straight i with hat on top space minus space 16 space straight j with hat on top space plus space 7 space straight k with hat on top close parentheses

and

straight r with rightwards arrow on top space equals space 15 space straight i with hat on top space plus space 29 space straight j with hat on top space plus space 5 space straight k with hat on top space plus space straight mu space open parentheses 3 space straight i with hat on top space plus space 8 space straight j with hat on top space minus space 5 space straight k with hat on top close parentheses

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 space stack straight b subscript 1 with rightwards arrow on top

and

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

, we get,

stack straight a subscript 1 with rightwards arrow on top space equals space 8 space straight i with hat on top space minus space 9 space straight j with hat on top space plus space 10 space straight k with hat on top comma space space stack straight b subscript 1 with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 16 space straight j with hat on top space plus space 7 space straight k with hat on top stack straight a subscript 2 with rightwards arrow on top space equals space 15 space straight i with hat on top space plus space 29 space straight j with hat on top space plus space 5 space straight k with hat on top comma space space space stack straight b subscript 1 with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 8 space straight j with hat on top space minus space 5 space straight k with hat on top stack straight a subscript 2 with rightwards arrow on top minus stack straight a subscript 1 with rightwards arrow on top space equals space 7 straight i with hat on top space plus space 38 space straight j with hat on top space minus space 5 space straight k with hat on top

Now

stack straight b subscript 1 with rightwards arrow on top cross times stack straight b subscript 2 with rightwards arrow on top space 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 3 cell negative 16 end cell 7 row 3 8 cell negative 5 end cell end table close vertical bar space equals space open vertical bar table row cell negative 16 end cell 7 row 8 cell negative 5 end cell end table close vertical bar space straight i with hat on top space minus space space open vertical bar table row 3 7 row 3 cell negative 5 end cell end table close vertical bar space straight j with hat on top space plus space open vertical bar table row 3 cell negative 16 end cell row 3 8 end table close vertical bar space straight k with hat on top

equals space left parenthesis 80 minus 56 right parenthesis space straight i with hat on top space minus space left parenthesis negative 15 minus 21 right parenthesis space straight j with hat on top space plus space left parenthesis 24 plus 48 right parenthesis space straight k with hat on top equals space 24 space straight i with hat on top space plus space 36 space straight j with hat on top space plus space 72 space straight k with hat on top space equals space 12 left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis

therefore

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 12 space square root of 4 plus 9 plus 36 end root space equals space 12 space cross times 7 space equals space 84

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 7 space straight i with hat on top space plus space 38 space straight j with hat on top space minus space 5 space straight k with hat on top right parenthesis. space 12 space left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 6 space straight k with hat on top right parenthesis over denominator 84 end fraction close vertical bar

equals space 1 over 7 left parenthesis 14 plus 114 minus 30 right parenthesis space equals space 1 over 7 left parenthesis 98 right parenthesis space equals space 14