Find the coordinates of the point where the line through A(3, 4,
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    Three Dimensional Geometry

    Multiple Choice QuestionsShort Answer Type

    51. Find a vector equation of the line through the points A (3, 4, – 7) and B (1, – 1, 6).
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    52. Find the vector equation for the line passing through the points (– 1, 0, 2) and (3, 4, 6).
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    53. Find the vector and cartesian equations of the line that passes through the origin and (5,- 2, 3).
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    54. Find the vector and the cartesian equations of the line that passes through the points (3, – 2, – 5), (3, – 2, 6).
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    55. Find the vector equation of line joining the points whose vectors are 2 space straight i with hat on top minus straight j with hat on top plus straight k with hat on top space space and space straight i with hat on top space plus space 2 space straight j with hat on top space minus space 3 space straight k with hat on top.
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    56. If the points (–1, 3, 2), (– 4, 2, –2) and (5, 5, λ) are collinear, then find the value of λ.
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    Multiple Choice QuestionsLong Answer Type

    57. Show that the points whose position vectors are given by 
    2 space straight i with hat on top space plus space straight j with hat on top space plus space 3 space straight k with hat on top comma space space minus space 4 space straight i with hat on top space plus space 3 space straight j with hat on top space minus space straight k with hat on top comma space space 5 space straight i with hat on top space plus space 5 space straight k with hat on top are collinear. 
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    Multiple Choice QuestionsShort Answer Type

    58.

    Show that the point whose position vectors are given by negative 2 space straight i with hat on top space plus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top comma space space straight i with hat on top space plus space 2 space straight j with hat on top space plus space 3 space straight k with hat on top space space and space space 7 straight i with hat on top space minus space straight k with hat on top are collinear. 

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    Multiple Choice QuestionsLong Answer Type

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    59. Find the coordinates of the point where the line through A(3, 4, 1) and B (5, 1, 6) crosses the x y-plane.

    The line passes through the points A (3, 4, 1) and B (5, 1, 6)
    therefore space space space space straight a with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus 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 space 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 space space space space straight b with rightwards arrow on top space minus space 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 5 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 thin space left parenthesis straight b with rightwards arrow on top minus straight a with rightwards arrow on top right parenthesis
    or              straight r with rightwards arrow on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space straight k with hat on top space plus space straight lambda space left parenthesis 2 space straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis                 ...(1)
    Let P (x, y, 0) be the point where the line AB crosses the x y-plane. Now P lies on (1)
    therefore space space space straight x space straight i with hat on top space plus space straight y space straight j with hat on top space space lies space on space left parenthesis 1 right parenthesis therefore space space space straight x space straight i with hat on top space plus space straight y space straight j with hat on top space equals space 3 space straight i with hat on top space plus space 4 space straight j with hat on top space plus space straight k with hat on top space plus space straight lambda left parenthesis 2 straight i with hat on top space minus space 3 space straight j with hat on top space plus space 5 space straight k with hat on top right parenthesis therefore space space space space straight x space straight i with hat on top space plus space straight y space straight j with hat on top space equals space left parenthesis 3 plus 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 plus space left parenthesis 1 plus 5 space straight lambda right parenthesis space straight k with hat on top space space space space space space space space space space space space space space space space space space space space space
    therefore                                 straight x space equals space 3 plus 2 space straight lambda                                                    ...(2)
                                         straight y space equals 4 minus 3 straight lambda                                                       ...(3)
                                          0 space equals space 1 plus 5 space straight lambda                                                    ...(4)
    From (4),   5 space straight lambda space equals space minus 1 space space space space space or space space space space space straight lambda space equals space minus 1 fifth
    From (2),        straight x space equals space 3 minus 2 over 5 space equals space 13 over 5
    From (3),     straight y space equals 4 plus 3 over 5 space equals space 23 over 5
    therefore point is  open parentheses 13 over 5 comma space 23 over 5 comma space 0 close parentheses
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    60. Find the coordinates of the point where the line through (5, 1, 6) and (3, 4, 1) crosses the YZ-plane.
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