Find the shortest distance between the lines:and from Mathema

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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

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

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 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 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

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 space space and space straight r with rightwards arrow on top space equals 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

we get

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

stack straight a subscript 2 with rightwards arrow on top space minus space straight a with rightwards arrow on top space equals space straight i with hat on top

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

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116. Find the shortest distance between the lines whose vector equations are:

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

and

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

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117. Find the shortest distance between the lines whose vector equations are

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

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118. Find the shortest distance between the following 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 straight k with hat on top space plus space straight lambda space left parenthesis straight i with hat on top space minus space straight j 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 minus 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 straight j with hat on top space plus space 2 space straight k with hat on top right parenthesis

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119. 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 plus space space 8 space straight j with hat on top space plus space 3 space straight k with hat on top space plus space straight lambda space left parenthesis 3 space straight i with hat on top space minus space straight j 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 minus 3 space straight i with hat on top space minus space 7 space straight j with hat on top space plus space 6 space straight k with hat on top space plus space straight mu space open parentheses negative 3 space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 4 space straight k with hat on top close parentheses space

.

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120. Find the shortest distance between the lines

straight r with rightwards arrow on top space equals space left parenthesis straight lambda minus 1 right parenthesis space straight i with hat on top space plus space left parenthesis straight lambda plus 1 right parenthesis space straight j with hat on top space minus space left parenthesis straight lambda plus 1 right parenthesis space straight k with hat on top

and

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

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