Find the vector and the Cartesian equations for the line through

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 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).
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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).


The line passes through the points A (3, 4, –7) and B (5, 1,6).
therefore space space space space space space space straight a with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 4 space straight j with hat on top space minus space 7 space straight k with hat on top comma space space space straight b with rightwards arrow on top space equals space 5 space straight i with hat on top space plus space straight j with hat on top space plus 6 space straight k with hat on top
where straight a with rightwards arrow on top comma space straight b with rightwards arrow on top are position vectors of A, B respectively.
therefore                            straight b with rightwards arrow on top minus straight a with rightwards arrow on top space equals space 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 13 space straight k with hat on top
Let straight r with rightwards arrow on top be the position vector of any point on the given line. 
∴   the vector equation of the line is
                  straight r with rightwards arrow on top space equals space straight a with rightwards arrow on top space plus space straight lambda left parenthesis straight b with rightwards arrow on top minus stack straight a right parenthesis with rightwards arrow on top
or              straight r with rightwards arrow on top space equals space 3 straight i with hat on top space plus space 4 straight j with hat on top space minus space 7 straight k with hat on top space plus space straight lambda left parenthesis 2 straight i with hat on top space minus space 3 straight j with hat on top space plus space 13 straight k with hat on top right parenthesis
Taking     straight r with rightwards arrow on top space equals space straight x space straight i with hat on top space plus space straight y space straight j with hat on top space plus space straight z space straight k with hat on top comma space space the space equation space of space line space is
straight x space straight i with hat on top space plus straight y space straight j with hat on top space space plus space straight z space straight k with hat on top space equals space left parenthesis 3 space plus space 2 space straight lambda right parenthesis space straight i with hat on top space plus space left parenthesis 4 minus 3 space straight lambda right parenthesis space straight j with hat on top space space plus space left parenthesis negative 7 plus 13 space straight lambda right parenthesis space straight k with hat on top
Comparing the coefficients of straight i with hat on top comma space straight j with hat on top comma space straight k with hat on top we get,
                           straight x space equals space 3 plus 2 space straight lambda comma space space space straight y space equals space 4 minus 3 straight lambda comma space space space straight z space equals space minus 7 space plus space 13 space straight lambda
rightwards double arrow space space space space space space space space fraction numerator straight x minus 3 over denominator 2 end fraction space equals space fraction numerator straight y minus 4 over denominator negative 3 end fraction space equals space fraction numerator straight z plus 7 over denominator 13 end fraction space left parenthesis space equals space straight lambda right parenthesis
which is the Cartesian equation of line.
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