Find the angle between the lines from Mathematics Three Dimensi
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    Three Dimensional Geometry

    Multiple Choice QuestionsShort Answer Type

    71. Find the angle between the lines whose direction ratios are a, b, c and b – c, c – a, a – b.
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    Answer
    72. Find the angle between each of the following pairs of lines:
    fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator negative 2 end fraction comma space space straight z space equals space 2 comma space space space fraction numerator straight x minus 1 over denominator 1 end fraction space equals space fraction numerator 2 straight y plus 3 over denominator 3 end fraction space equals space fraction numerator straight z plus 5 over denominator 2 end fraction
    84 Views
    Answer
    73. Find the angle between pair of lines:
                       fraction numerator straight x plus 3 over denominator 3 end fraction space equals space fraction numerator straight y minus 1 over denominator 5 end fraction space equals space fraction numerator straight z plus 3 over denominator 4 end fraction

    and          fraction numerator straight x plus 1 over denominator 1 end fraction space equals space fraction numerator straight y minus 4 over denominator 1 end fraction space equals space fraction numerator straight z minus 5 over denominator 2 end fraction
    79 Views
    Answer
    74. Find the angle between the following pair of lines:
    fraction numerator straight x minus 2 over denominator 2 end fraction space equals space fraction numerator straight y minus 1 over denominator 5 end fraction space equals fraction numerator straight z plus 3 over denominator negative 3 end fraction space space and space space fraction numerator straight x plus 1 over denominator negative 1 end fraction space equals space fraction numerator straight y minus 4 over denominator 8 end fraction space equals space fraction numerator straight z minus 5 over denominator 4 end fraction
    95 Views
    Answer
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    75. Find the angle between the following pair of lines:
    fraction numerator 5 minus straight x over denominator 3 end fraction space equals space fraction numerator straight y plus 3 over denominator negative 2 end fraction comma space space straight z space equals space 5 space space space and space space straight x over 1 space equals space fraction numerator 1 minus straight y over denominator 3 end fraction space equals fraction numerator straight z minus 5 over denominator 2 end fraction
    77 Views
    Answer
    76. Find the angle between the following pair of lines:
     straight x over 2 space equals space straight y over 2 space equals space straight z over 1 space space and space space fraction numerator straight x minus 5 over denominator 4 end fraction space equals space fraction numerator straight y minus 2 over denominator 1 end fraction space equals space fraction numerator straight z minus 3 over denominator 8 end fraction
    87 Views
    Answer
    77. Find the angle between the following pair of lines:
    fraction numerator straight x minus 5 over denominator 1 end fraction equals space fraction numerator 2 straight y plus 6 over denominator negative 2 end fraction space equals space fraction numerator straight z minus 3 over denominator 1 end fraction space space and space fraction numerator straight x minus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator 4 end fraction space equals space fraction numerator straight z minus 6 over denominator 5 end fraction
     
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    78. Find the angle between the lines
    straight x over 1 space equals fraction numerator negative straight y over denominator 0 end fraction space equals space fraction numerator straight z over denominator negative 1 end fraction space space and space space straight x over 3 space equals space straight y over 4 space equals straight z over 5

    The given lines are
    straight x over 1 space equals space straight y over 0 space equals space fraction numerator straight z over denominator negative 1 end fraction space space space and space space straight x over 3 space equals space straight y over 4 space equals space straight z over 5
    Let  stack straight b subscript 1 with rightwards arrow on top comma space stack straight b subscript 2 with rightwards arrow on top be vectors parallel to these lines
    therefore 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 minus space straight k with hat on top comma space space stack straight b 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 plus space 5 space straight k with hat on top
    Let θ be the angle between given lines.
    ∴   θ is angle between stack straight b subscript 1 with rightwards arrow on top space space and space space stack straight b subscript 2 with rightwards arrow on top
    therefore space space space space space space space cos space straight theta space equals space fraction numerator stack straight b subscript 1 with rightwards arrow on top. space stack straight b subscript 2 with rightwards arrow on top over denominator open vertical bar stack straight b subscript 1 with rightwards arrow on top close vertical bar space open vertical bar stack straight b subscript 2 with rightwards arrow on top close vertical bar end fraction space equals space fraction numerator left parenthesis 1 right parenthesis thin space left parenthesis 3 right parenthesis space plus space left parenthesis 0 right parenthesis thin space left parenthesis 4 right parenthesis space plus space left parenthesis negative 1 right parenthesis thin space left parenthesis 5 right parenthesis over denominator square root of 1 plus 0 plus 1 end root space square root of 9 plus 16 plus 25 end root end fraction space space space space space space space space space space space space space space space space space space space space space space equals space fraction numerator 3 minus 5 over denominator square root of 2 space square root of 50 end fraction space equals space fraction numerator negative 2 over denominator square root of 2 cross times 5 square root of 2 end fraction space equals space minus 1 fifth
    ∴   acute angle θ is given by
             cos space straight theta space equals space 1 fifth
    therefore space space space space space straight theta space equals cos to the power of negative 1 end exponent open parentheses 1 fifth close parentheses
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    Multiple Choice QuestionsLong Answer Type

    79. Find the value of p so that the lines
    fraction numerator 1 minus straight x over denominator 3 end fraction space equals space fraction numerator 7 straight y minus 14 over denominator 2 space straight p end fraction space equals space fraction numerator straight z minus 3 over denominator 2 end fraction space space and space space fraction numerator 7 minus 7 straight z over denominator 3 space straight p end fraction space equals space fraction numerator straight y minus 5 over denominator 1 end fraction space equals space fraction numerator 6 minus straight z over denominator 5 end fraction
    80 Views
    Answer

    Multiple Choice QuestionsShort Answer Type

    80. Show that the lines
    fraction numerator straight x minus 5 over denominator 7 end fraction space equals space fraction numerator straight y plus 2 over denominator negative 5 end fraction space equals straight z over 1 space and space straight x over 1 space equals space straight y over 2 space equals space straight z over 3 are prependicular to each other. 
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    Answer
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