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Three Dimensional Geometry

Question
CBSEENMA12033257

Find the vector equation of the line passing through the point (2, –1, –1) which is parallel to the lines 6 x – 2 = 3 y + 1 = 2 z – 2.

Solution

Here straight a with rightwards arrow on top space equals space 2 straight i with overparenthesis on top space minus straight j with overparenthesis on top space minus straight k with overparenthesis on top
The equation of line is
           6 straight x minus 2 space equals space 3 straight y plus 1 space equals space 2 straight z minus 2
or       fraction numerator straight x minus begin display style 1 third end style over denominator begin display style 1 over 6 end style end fraction space equals space fraction numerator straight y plus begin display style 1 third end style over denominator begin display style 1 third end style end fraction space equals space fraction numerator straight z minus 1 over denominator begin display style 1 half end style end fraction space space space or space fraction numerator straight x minus begin display style 1 third end style over denominator 1 end fraction space equals fraction numerator straight y plus begin display style 1 third end style over denominator 2 end fraction space equals space fraction numerator straight z minus 1 over denominator 3 end fraction
∴     direction ratios of the line are 1, 2, 3.
∴     line is parallel to the vector
straight b with rightwards arrow on top space equals space straight i with overparenthesis on top space plus space 2 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 left parenthesis 2 straight i with overparenthesis on top minus straight j with overparenthesis on top plus straight k with overparenthesis on top right parenthesis space plus space straight lambda left parenthesis straight i with overparenthesis on top plus 2 space straight j with hat on top space plus space 3 space straight k with hat on top right parenthesis