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Three Dimensional Geometry

Question

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

Solution

The equations of the two lines 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 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 space straight j with hat on top space minus space left parenthesis 2 straight s plus 1 right parenthesis space straight k with hat on top

or

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

...(1)
and

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

...(2)
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 t space space stack straight b subscript 1 with rightwards arrow on top space space space and space 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 s space stack straight b subscript 2 with rightwards arrow on top stack straight a subscript 1 with rightwards arrow on top space equals space straight i with hat on top space minus space 2 space space straight j with hat on top space plus space 3 space straight k with hat on top comma space space space stack straight b subscript 1 with rightwards arrow on top space equals space minus space straight i with hat on top space plus space straight j with hat on top space minus space 2 space straight k with hat on top stack straight a subscript 2 with rightwards arrow on top space space equals straight i with hat on top space minus space straight j 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 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

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 straight j with hat on top space minus space 4 space straight k with hat on top

Now,

stack straight b subscript 1 with rightwards arrow on top space cross times space 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 cell negative 1 end cell 1 cell negative 2 end cell row 1 2 cell negative 2 end cell end table close vertical bar

equals space left parenthesis negative 2 plus 4 right parenthesis space straight i with hat on top space minus space left parenthesis 2 plus 2 right parenthesis space straight j with hat on top space space plus space left parenthesis negative 2 minus 1 right parenthesis space straight k with hat on top space equals space 2 space straight i with hat on top space minus space 4 space straight j with hat on top space minus space 3 space straight k with hat on top

therefore space space space 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 square root of 4 plus 16 plus 9 end root space equals square root of 29

Let

PQ with rightwards arrow on top

be the S.D. vector between the 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.

If

straight n with rightwards arrow on top

is a unit vector along

PQ with rightwards arrow on top

,
then

straight n with rightwards arrow on top space equals space fraction numerator stack straight b subscript 1 with rightwards arrow on top 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 fraction numerator 1 over denominator square root of 29 end fraction left parenthesis 2 space straight i with hat on top space minus space 4 space straight j with hat on top space minus 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 on 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

equals space left parenthesis straight j with hat on top space minus space 4 space straight k with hat on top right parenthesis. space fraction numerator 1 over denominator square root of 29 end fraction left parenthesis 2 space straight i with hat on top space minus space 4 space space straight j with hat on top space minus space space 3 space straight k with hat on top right parenthesis

equals space fraction numerator 1 over denominator square root of 29 end fraction left parenthesis 0 minus 4 plus 12 right parenthesis space equals space fraction numerator 8 over denominator square root of 29 end fraction space units. space

Some More Questions From Three Dimensional Geometry Chapter

Find the direction cosines of x, y and z-axis.

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