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The given line is <img class=Wirisformula style=vertical-align: -9px; src=/application/zrc/images/qvar/MAEN12064338-1.png alt=fraction numerator negative straight x minus 2 over denominator 1 end fraction space equals space fraction numerator straight y plus 3 over denominator 7 end fraction space equals space fraction numerator 2 straight z minus 6 over denominator 3 end fraction width=166 height=25 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mo»-«/mo»«mi mathvariant=¨normal¨»x«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«mn»1«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»y«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«mn»7«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mrow»«mn»2«/mn»«mi mathvariant=¨normal¨»z«/mi»«mo»-«/mo»«mn»6«/mn»«/mrow»«mn»3«/mn»«/mfrac»«/math»> or <img class=Wirisformula style=vertical-align: -25px; src=/application/zrc/images/qvar/MAEN12064338-2.png alt=fraction numerator straight x plus 2 over denominator negative 1 end fraction space equals space fraction numerator straight y plus 3 over denominator 7 end fraction space equals space fraction numerator straight z minus 3 over denominator begin display style 3 over 2 end style end fraction space space space space space space space or space space space space space fraction numerator straight x plus 2 over denominator negative 2 end fraction space equals space fraction numerator straight y plus 3 over denominator 14 end fraction space equals fraction numerator straight z minus 3 over denominator 3 end fraction width=339 height=41 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi mathvariant=¨normal¨»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«mrow»«mo»-«/mo»«mn»1«/mn»«/mrow»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»y«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«mn»7«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»z«/mi»«mo»-«/mo»«mn»3«/mn»«/mrow»«mstyle displaystyle=¨true¨»«mfrac»«mn»3«/mn»«mn»2«/mn»«/mfrac»«/mstyle»«/mfrac»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mi»or«/mi»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»x«/mi»«mo»+«/mo»«mn»2«/mn»«/mrow»«mrow»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»y«/mi»«mo»+«/mo»«mn»3«/mn»«/mrow»«mn»14«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»z«/mi»«mo»-«/mo»«mn»3«/mn»«/mrow»«mn»3«/mn»«/mfrac»«/math»> Its direction ratios are –2, 14, 3 ∴ cartesian equation of line through (1, 2, 3) and having direction ratios –2, 14, 3 are <img class=Wirisformula style=vertical-align: -9px; src=/application/zrc/images/qvar/MAEN12064338-3.png alt=fraction numerator straight x minus 1 over denominator negative 2 end fraction space equals space fraction numerator straight y minus 2 over denominator 14 end fraction space equals fraction numerator straight z minus 3 over denominator 3 end fraction width=144 height=25 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mfrac»«mrow»«mi mathvariant=¨normal¨»x«/mi»«mo»-«/mo»«mn»1«/mn»«/mrow»«mrow»«mo»-«/mo»«mn»2«/mn»«/mrow»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»y«/mi»«mo»-«/mo»«mn»2«/mn»«/mrow»«mn»14«/mn»«/mfrac»«mo»§#160;«/mo»«mo»=«/mo»«mfrac»«mrow»«mi mathvariant=¨normal¨»z«/mi»«mo»-«/mo»«mn»3«/mn»«/mrow»«mn»3«/mn»«/mfrac»«/math»> Now, <img class=Wirisformula style=vertical-align: -2px; src=/application/zrc/images/qvar/MAEN12064338-4.png alt=straight a with rightwards arrow on top space equals space straight i with overparenthesis on top space plus space 2 straight j with overparenthesis on top plus 3 stack straight k comma with overparenthesis on top space space space straight m with rightwards arrow on top space equals space minus 2 space straight i with overparenthesis on top space plus space 14 space straight j with hat on top space plus space 3 space straight k with hat on top width=255 height=17 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi mathvariant=¨normal¨»a«/mi»«mo»§#8594;«/mo»«/mover»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mover»«mi mathvariant=¨normal¨»i«/mi»«mo»§#9180;«/mo»«/mover»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mn»2«/mn»«mover»«mi mathvariant=¨normal¨»j«/mi»«mo»§#9180;«/mo»«/mover»«mo»+«/mo»«mn»3«/mn»«mover»«mrow»«mi mathvariant=¨normal¨»k«/mi»«mo»,«/mo»«/mrow»«mo»§#9180;«/mo»«/mover»«mo»§#160;«/mo»«mo»§#160;«/mo»«mo»§#160;«/mo»«mover»«mi mathvariant=¨normal¨»m«/mi»«mo»§#8594;«/mo»«/mover»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»-«/mo»«mn»2«/mn»«mo»§#160;«/mo»«mover»«mi mathvariant=¨normal¨»i«/mi»«mo»§#9180;«/mo»«/mover»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mn»14«/mn»«mo»§#160;«/mo»«mover»«mi mathvariant=¨normal¨»j«/mi»«mo»^«/mo»«/mover»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mover»«mi mathvariant=¨normal¨»k«/mi»«mo»^«/mo»«/mover»«/math»> ∴ vector equation of line is <img class=Wirisformula style=vertical-align: -1px; src=/application/zrc/images/qvar/MAEN12064338-5.png alt=straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top plus straight lambda straight m with rightwards arrow on top width=69 height=13 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi mathvariant=¨normal¨»r«/mi»«mo»§#8594;«/mo»«/mover»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mover»«mi mathvariant=¨normal¨»a«/mi»«mo»§#8594;«/mo»«/mover»«mo»+«/mo»«mi mathvariant=¨normal¨»§#955;«/mi»«mover»«mi mathvariant=¨normal¨»m«/mi»«mo»§#8594;«/mo»«/mover»«/math»> or <img class=Wirisformula style=vertical-align: -2px; src=/application/zrc/images/qvar/MAEN12064338-6.png alt=straight r with rightwards arrow on top space equals space left parenthesis straight i with hat on top plus 2 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis plus straight lambda left parenthesis negative 2 space straight i with hat on top plus 14 space straight j with hat on top plus 3 space straight k with hat on top right parenthesis. width=241 height=17 align=middle data-mathml=«math xmlns=¨http://www.w3.org/1998/Math/MathML¨»«mover»«mi mathvariant=¨normal¨»r«/mi»«mo»§#8594;«/mo»«/mover»«mo»§#160;«/mo»«mo»=«/mo»«mo»§#160;«/mo»«mo»(«/mo»«mover»«mi mathvariant=¨normal¨»i«/mi»«mo»^«/mo»«/mover»«mo»+«/mo»«mn»2«/mn»«mo»§#160;«/mo»«mover»«mi mathvariant=¨normal¨»j«/mi»«mo»^«/mo»«/mover»«mo»§#160;«/mo»«mo»+«/mo»«mo»§#160;«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mover»«mi mathvariant=¨normal¨»k«/mi»«mo»^«/mo»«/mover»«mo»)«/mo»«mo»+«/mo»«mi mathvariant=¨normal¨»§#955;«/mi»«mo»(«/mo»«mo»-«/mo»«mn»2«/mn»«mo»§#160;«/mo»«mover»«mi mathvariant=¨normal¨»i«/mi»«mo»^«/mo»«/mover»«mo»+«/mo»«mn»14«/mn»«mo»§#160;«/mo»«mover»«mi mathvariant=¨normal¨»j«/mi»«mo»^«/mo»«/mover»«mo»+«/mo»«mn»3«/mn»«mo»§#160;«/mo»«mover»«mi mathvariant=¨normal¨»k«/mi»«mo»^«/mo»«/mover»«mo»)«/mo»«mo».«/mo»«/math»>

For the cartesian and vector equation of a line which passes through the point (1, 2, 3) and is parallel to the line $fraction numerator negative straight x minus 2 over denominator 1 end fraction space equals space fraction numerator straight y plus 3 over denominator 7 end fraction space equals space fraction numerator 2 straight z minus 6 over denominator 3 end fraction.$

The given line is

fraction numerator negative straight x minus 2 over denominator 1 end fraction space equals space fraction numerator straight y plus 3 over denominator 7 end fraction space equals space fraction numerator 2 straight z minus 6 over denominator 3 end fraction

fraction numerator straight x plus 2 over denominator negative 1 end fraction space equals space fraction numerator straight y plus 3 over denominator 7 end fraction space equals space fraction numerator straight z minus 3 over denominator begin display style 3 over 2 end style end fraction space space space space space space space or space space space space space fraction numerator straight x plus 2 over denominator negative 2 end fraction space equals space fraction numerator straight y plus 3 over denominator 14 end fraction space equals fraction numerator straight z minus 3 over denominator 3 end fraction

Its direction ratios are –2, 14, 3
∴ cartesian equation of line through (1, 2, 3) and having direction ratios –2, 14, 3 are

fraction numerator straight x minus 1 over denominator negative 2 end fraction space equals space fraction numerator straight y minus 2 over denominator 14 end fraction space equals fraction numerator straight z minus 3 over denominator 3 end fraction

Now,

straight a with rightwards arrow on top space equals space straight i with overparenthesis on top space plus space 2 straight j with overparenthesis on top plus 3 stack straight k comma with overparenthesis on top space space space straight m with rightwards arrow on top space equals space minus 2 space straight i with overparenthesis on top space plus space 14 space straight j with hat on top space plus space 3 space straight k with hat on top

∴ vector equation of line is

straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top plus straight lambda straight m with rightwards arrow on top

straight r with rightwards arrow on top space equals space left parenthesis straight i with hat on top plus 2 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis plus straight lambda left parenthesis negative 2 space straight i with hat on top plus 14 space straight j with hat on top plus 3 space straight k with hat on top right parenthesis.

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