Find the equation of a line, which is parallel to  and passes
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    Three Dimensional Geometry

    Multiple Choice QuestionsShort Answer Type

    41. The equation of a line is fraction numerator 2 straight x minus 5 over denominator 4 end fraction space equals space fraction numerator straight y plus 4 over denominator 3 end fraction space equals fraction numerator 6 minus straight z over denominator 6 end fraction. Find the direction cosines of a line parallel to this line. 
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    Multiple Choice QuestionsLong Answer Type

    42. The cartesian equations of a line are 3 x + 1 = 6 y – 2 = 1 – z. Find the fixed point through which it passes, its direction ratios and also its vector equation.
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    Multiple Choice QuestionsShort Answer Type

    43. Find the equation of the line which passes through the point (3, 4, 5) and is parallel to the vector 2 straight i with hat on top plus 2 straight j with hat on top minus 3 straight k with hat on top.
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    44. Find the vector and Cartesian equations of the line through the point (5, 2, – 4) and which is parallel to the vector 3 straight i with overparenthesis on top space plus space 2 straight j with overparenthesis on top space minus space 8 space straight k with overparenthesis on top.
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    45.

    Find the equation of the line which passes through the point (1, 2, 3) and is parallel to the vector 3 straight i with overparenthesis on top space plus space 2 straight j with overparenthesis on top space minus space 2 straight k with overparenthesis on top.

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    46. Find equation of line, which is parallel to 2 straight i with hat on top plus straight j with hat on top minus straight k with hat on top and passes through (1, 2, 3).
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    47. Find the equation of a line, which is parallel to straight i with hat on top plus straight j with hat on top plus straight k with hat on top and passes through (2, 3, 1).

    We know that the equation of a straight line passing through a fixed point with position vector straight a with rightwards arrow on top and parallel to the vector straight m with rightwards arrow on top space is space 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
    where λ is a parameter.
    Here straight a with rightwards arrow on top space equals space 2 straight i with hat on top space plus space 3 straight j with hat on top space plus space straight k with hat on top space and space straight m with rightwards arrow on top space equals space straight i with hat on top plus straight j with hat on top plus straight k with hat on top
    ∴    from (1), the vector equation of line is
    straight r with rightwards arrow on top space equals space left parenthesis 2 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 space plus space straight lambda space left parenthesis straight i with hat on top space plus space straight j with hat on top space plus space straight k with hat on top right parenthesis
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    48. Find equation of a line, which is parallel to 2 straight i with hat on top space plus space straight j with hat on top space minus space 3 space straight k with hat on top and passes through (1, ,1, 1).
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    Multiple Choice QuestionsLong Answer Type

    49. Find the point on the line fraction numerator straight x plus 2 over denominator 3 end fraction space equals space fraction numerator straight y plus 1 over denominator 2 end fraction space equals space fraction numerator straight z minus 3 over denominator 2 end fraction at a distance 3 square root of 2 from the point (1, 2, 3).
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    50. Find the vector and the Cartesian equations for the line through the points A (3, 4, –7) and B (5, 1, 6).
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