Find the shortest distance between the lines: from Mathematics

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Three Dimensional Geometry

Short Answer Type

121. Find the shortest distance between the lines:

straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space minus space 15 space straight j with hat on top space plus space 9 space straight k with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space minus space 7 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals space left parenthesis 2 space straight mu minus 1 right parenthesis space straight i with hat on top space plus space left parenthesis 1 plus straight mu right parenthesis space straight j with hat on top space plus space left parenthesis 9 minus 3 space straight mu right parenthesis space straight k with hat on top

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Long Answer Type

122. Find the shortest distance between the lines:

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

and

straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 4 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 4 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses.

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123. Find the shortest distance between the lines:

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

and

straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 4 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 4 space straight j with hat on top space plus space 5 space straight k with hat on top close parentheses.

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124. 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

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125. Find the shortest distance between the lines:

straight r with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space straight lambda space left parenthesis straight i with hat on top space plus space 2 space straight j with hat on top space minus space 2 space straight k with hat on top right parenthesis

and

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

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126. 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.

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Short Answer Type

127. Find the distance between the lines l₁, and l₂ given by

straight r with rightwards arrow on top space equals space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space space 6 space straight k with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 3 straight j with hat on top space minus space 5 space straight k with hat on top space plus space straight mu 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

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Long Answer Type

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128. Find the shortest distance between the lines:
$fraction numerator straight x plus 1 over denominator 7 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 6 end fraction space equals space fraction numerator straight z plus 1 over denominator 1 end fraction space and space fraction numerator straight x minus 3 over denominator 1 end fraction space equals space fraction numerator straight y minus 5 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 7 over denominator 1 end fraction$

The equations of two lines are

fraction numerator straight x plus 1 over denominator 7 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 6 end fraction space equals space fraction numerator straight z plus 1 over denominator 1 end fraction space and space fraction numerator straight x minus 3 over denominator 1 end fraction space equals space fraction numerator straight y minus 5 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 7 over denominator 1 end fraction

or

fraction numerator straight x minus left parenthesis negative 1 right parenthesis over denominator 7 end fraction space equals space fraction numerator straight y minus left parenthesis negative 1 right parenthesis over denominator negative 6 end fraction space equals space fraction numerator straight z minus left parenthesis negative 1 right parenthesis over denominator 1 end fraction space and space fraction numerator straight x minus 3 over denominator 1 end fraction space equals space fraction numerator straight y minus 5 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 7 over denominator 1 end fraction

therefore

straight x subscript 1 space equals space minus 1 comma space space space straight y subscript 1 space equals space minus 1 comma space space straight z subscript 1 space equals space minus 1 semicolon space space straight a subscript 1 space equals space 7 comma space space straight b subscript 1 space equals space minus 6 comma space space straight c subscript 1 space equals space 1

and

straight x subscript 2 space equals space 3 comma space space space straight y subscript 2 equals 5 comma space space space straight z subscript 2 space equals space 7 semicolon space space space straight a subscript 2 space equals space 1 comma space space straight b subscript 2 space equals space minus 2 comma space space straight c subscript 2 space equals 1

therefore space space space space straight x subscript 2 minus straight x subscript 1 space equals space 4 comma space space space space space straight y subscript 2 minus straight y subscript 1 space equals space 6 comma space space space straight z subscript 2 space minus space straight z subscript 1 space equals space 8

therefore space space space space open vertical bar table row cell straight x subscript 2 minus straight x subscript 1 end cell cell straight y subscript 2 minus straight y subscript 1 end cell cell straight z subscript 2 minus straight z subscript 1 end cell row cell straight a subscript 1 end cell cell straight b subscript 1 end cell cell straight c subscript 1 end cell row cell straight a subscript 2 end cell cell straight b subscript 2 end cell cell straight c subscript 2 end cell end table close vertical bar space equals space open vertical bar table row 4 6 8 row 7 cell negative 6 end cell 1 row 1 cell negative 2 end cell 1 end table close vertical bar

equals space 4 space open vertical bar table row cell negative 6 end cell 1 row cell negative 2 end cell 1 end table close vertical bar space minus space 6 space open vertical bar table row 7 1 row 1 1 end table close vertical bar space plus space 8 space open vertical bar table row 7 cell negative 6 end cell row 1 cell negative 2 end cell end table close vertical bar

equals space 4 space left parenthesis negative 6 plus 2 right parenthesis space minus space 6 space left parenthesis 7 minus 1 right parenthesis space plus space 8 thin space left parenthesis negative 14 plus 6 right parenthesis equals space 4 left parenthesis negative 4 right parenthesis space minus space 6 space left parenthesis 6 right parenthesis space plus space 8 space left parenthesis negative 8 right parenthesis equals space minus 16 minus 36 minus 64 space equals space minus 116

square root of left parenthesis straight b subscript 1 straight c subscript 2 minus straight b subscript 2 straight c subscript 1 right parenthesis squared plus left parenthesis straight c subscript 1 straight a subscript 2 minus straight c subscript 2 straight a subscript 1 right parenthesis squared plus left parenthesis straight a subscript 1 straight b subscript 2 minus straight a subscript 2 straight b subscript 1 right parenthesis squared end root

equals space square root of left parenthesis negative 6 plus 2 right parenthesis squared plus left parenthesis 1 minus 7 right parenthesis squared plus left parenthesis negative 14 plus 6 right parenthesis squared end root space equals space square root of 16 plus 36 plus 64 end root space equals space square root of 116

therefore space space space straight S. straight D. space equals space open vertical bar fraction numerator negative 116 over denominator square root of 116 end fraction close vertical bar space equals fraction numerator 116 over denominator square root of 116 end fraction space equals square root of 116 space equals space square root of 4 cross times 29 end root space equals space 2 square root of 29.

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129. Determine whether the following pair of lines intersect:

straight r with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space plus space straight lambda left parenthesis 2 space straight i with hat on top space plus space straight j with hat on top right parenthesis

and

straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight mu left parenthesis straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis

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130. Determine whether the following pair of lines intersect:

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

and

straight r with rightwards arrow on top space equals space 4 space straight i with hat on top space minus space straight k with hat on top space plus space straight mu space left parenthesis 2 straight i with hat on top space plus space 3 space stack straight k right parenthesis with hat on top

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