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Three Dimensional Geometry

Long Answer Type

111. Show that the st. lines whose direction-cosines are given by the equations u I + v m + w n = 0,f m n + g n l + h l m = 0 are
(i) perpendicular if

straight f over straight u plus straight g over straight v plus straight h over straight w equals space 0

(ii) parallel if

square root of straight u space straight f space end root space plus space square root of straight v space straight g end root space plus space square root of straight w space straight h end root space space equals space 0

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112.

Find the shortest distance (S.D.) between the lines:
$stack straight r subscript 1 with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 4 space straight k with hat on top space plus space straight lambda space left parenthesis 2 space straight i with hat on top space minus space straight j with hat on top space plus space stack straight k right parenthesis with hat on top stack straight r subscript 2 with rightwards arrow on top space space equals space 3 space straight i with hat on top space minus space 2 space straight j with hat on top space minus space straight k with hat on top space plus space straight mu space left parenthesis 3 space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis.$

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113. Find the shortest distance between the lines whose vector equations are
$straight r with rightwards arrow on top space equals left parenthesis space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space plus space straight lambda left parenthesis straight i with hat on top space minus space 3 space straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis$
and $straight r with rightwards arrow on top space equals space 4 straight i with hat on top space plus 5 space straight j with hat on top space plus space 6 space straight k with hat on top space plus space straight mu space left parenthesis 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space straight k with hat on top right parenthesis$

The equations of two lines are

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

...(1)
and

straight r with rightwards arrow on top space equals space 4 straight i with hat on top space plus 5 space straight j with hat on top space plus space 6 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 straight k with hat on top right parenthesis

...(2)
Comparing these equations with

straight r with rightwards arrow on top space equals 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 space space space and space 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 plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top comma space space stack straight b subscript 1 with rightwards arrow on top space equals space straight i with hat on top space minus 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 4 space straight i with hat on top space plus space 5 space straight j with hat on top space plus space 6 space straight k with hat on top comma space space space stack straight b subscript 2 with rightwards arrow on top space equals space 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space straight k with hat on top

Let S be point on line (1) with position vector

stack straight a subscript 1 with rightwards arrow on top

and T be point on line (2) with position vector

stack straight a subscript 2 with rightwards arrow on top

so that

ST with rightwards arrow on top space equals space 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 3 space straight i with hat on top space plus space space 3 space straight j with hat on top space plus space 3 space straight k with hat on top

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

equals space left parenthesis negative 3 minus 6 right parenthesis space straight i with hat on top minus left parenthesis 1 minus 4 right parenthesis space straight j with hat on top space plus left parenthesis 3 plus 6 right parenthesis space straight k with hat on top space equals space minus 9 straight i with hat on top plus space 3 straight j with hat on top space plus space 9 straight k with hat on top

open vertical bar stack straight b subscript 1 with rightwards arrow on top cross times space stack straight b subscript 2 with rightwards arrow on top close vertical bar space equals space square root of 81 plus 9 plus 81 end root space equals space square root of 171

Let

PQ with rightwards arrow on top

be the S.D. vector between given lines.
Therefore, it is parallel to

stack straight b subscript 1 with rightwards arrow on top space cross times space stack straight b subscript 2 with rightwards arrow on top.

straight n with rightwards arrow on top

is a unit vector along

PQ with rightwards arrow on top comma

then

straight n with rightwards arrow on top space equals space fraction numerator stack straight b subscript 1 with rightwards arrow on top space cross times 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 cross times stack straight b subscript 2 with rightwards arrow on top close vertical bar end fraction space equals space minus fraction numerator 1 over denominator square root of 171 end fraction left parenthesis negative 9 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 9 space straight k with hat on top right parenthesis

equals space fraction numerator 1 over denominator square root of 19 end fraction left parenthesis negative 3 space straight i with hat on top space space plus space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis

Now S.D. = Projection of

ST with rightwards arrow on top space on space PQ with rightwards arrow on top

space equals space Projection space of space ST with rightwards arrow on top space space on space space straight n with rightwards arrow on top space equals space ST with rightwards arrow on top. space straight n with rightwards arrow on top space equals space left parenthesis 3 straight i with hat on top space plus space 3 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space. space fraction numerator 1 over denominator square root of 19 end fraction left parenthesis negative 3 space straight i with hat on top space plus space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis space equals space fraction numerator 1 over denominator square root of 19 end fraction open square brackets left parenthesis 3 right parenthesis space left parenthesis negative 3 right parenthesis space plus space left parenthesis 3 right parenthesis thin space left parenthesis 1 right parenthesis space plus space left parenthesis 3 right parenthesis thin space left parenthesis 3 right parenthesis close square brackets space equals space fraction numerator 1 over denominator square root of 19 end fraction space left square bracket negative 9 plus 3 plus 9 right square bracket space equals space fraction numerator 3 over denominator square root of 19 end fraction

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114. Find the shortest distance between the lines

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

and

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

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115.

Find the shortest distance between the lines:
$straight r with rightwards arrow on top space equals space straight i with hat on top space minus space straight j with hat on top space plus space straight lambda left parenthesis 2 straight i with hat on top space plus space straight k with hat on top right parenthesis$
and $straight r with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space straight j with hat on top space plus space straight mu left parenthesis straight i with hat on top space plus space straight j with hat on top space minus space straight k with hat on top right parenthesis$