Question

Show that lines:
$straight r with rightwards arrow on top space equals space straight i with hat on top space plus straight j with hat on top space plus straight k with hat on top space plus space straight lambda open parentheses straight i with hat on top minus straight j with hat on top space plus space straight k with hat on top close parentheses straight r with rightwards arrow on top space equals space 4 straight j with hat on top space plus space 2 straight k with hat on top space plus space straight mu open parentheses 2 straight j with hat on top minus straight j with hat on top plus 3 straight k with hat on top close parentheses space are space coplanar.$
Also, find the equation of the plane containing these lines.

Solution

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 k with hat on top space plus space straight lambda left parenthesis straight i with hat on top minus straight j with hat on top plus straight k with hat on top right parenthesis space... left parenthesis straight i right parenthesis

Converting into cartesian form,

fraction numerator straight x minus 1 over denominator 1 end fraction space equals fraction numerator straight y minus 1 over denominator negative 1 end fraction equals fraction numerator straight z minus 1 over denominator 1 end fraction left parenthesis straight x subscript 1 comma space straight y subscript 1 comma space straight z subscript 1 right parenthesis space equals space left parenthesis 1 comma space 1 comma space 1 right parenthesis straight a subscript 1 space equals space 1 comma space space straight b subscript 1 space equals negative 1 comma space space straight c subscript 1 space equals 1

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

Converting into cartesian form,

fraction numerator straight x minus 0 over denominator 2 end fraction equals fraction numerator straight y minus 4 over denominator negative 1 end fraction equals fraction numerator straight z minus 2 over denominator 3 end fraction left parenthesis straight x subscript 2 comma space straight y subscript 2 comma space straight z subscript 2 right parenthesis space equals space left parenthesis 0 comma space 4 comma space 2 right parenthesis straight a subscript 2 equals space 2 comma space space straight b subscript 2 equals negative 1 comma space space straight c subscript 2 equals 3

Condition for the lines to be coplanar is

open vertical bar table row cell 0 minus 1 end cell cell space 4 minus 1 end cell cell space 2 minus 1 end cell row 1 cell negative 1 end cell 1 row 2 cell negative 1 end cell 3 end table close vertical bar space equals space open vertical bar table row cell negative 1 end cell cell space space 3 end cell cell space space 1 end cell row 1 cell negative 1 end cell cell space space 1 end cell row 2 cell negative 1 end cell cell space space 3 end cell end table close vertical bar equals 0

therefore space the space lines space are space coplanar. Intersection space of space the space two space lines space is Let space the space equation space be space straight a left parenthesis straight x minus straight x subscript 1 right parenthesis plus straight b left parenthesis straight y minus straight y subscript 1 right parenthesis plus straight c left parenthesis straight z minus straight z subscript 1 right parenthesis space equals space 0 space... left parenthesis iii right parenthesis Direction space ratios space of space the space plane space is straight a minus straight b plus straight c equals 0 2 straight a minus straight b plus 3 straight c space equals 0 Solving space by space cross minus multiplication comma fraction numerator straight a over denominator negative 3 plus 1 end fraction equals fraction numerator straight b over denominator 2 minus 3 end fraction equals fraction numerator straight c over denominator negative 1 plus 2 end fraction straight a equals negative 2 straight lambda comma space space straight b equals negative straight lambda comma space space straight c equals straight lambda

Since the plane passes through (0, 4, 2) from line (ii)

straight a left parenthesis straight x minus 0 right parenthesis plus straight b left parenthesis straight y minus 4 right parenthesis plus straight c left parenthesis straight z minus 2 right parenthesis space equals 0 rightwards double arrow space minus 2 λx minus straight lambda left parenthesis straight y minus 4 right parenthesis plus straight lambda left parenthesis straight z minus 2 right parenthesis space equals 0 rightwards double arrow space minus 2 straight x minus straight y plus 4 plus straight z minus 2 equals 0 rightwards double arrow negative 2 straight x minus straight y plus straight z equals negative 2 rightwards double arrow 2 straight x plus straight y minus straight z equals 2

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