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Finding vector coordinates help

  1. Jun 8, 2008 #1
    My question is regarding the following problem:

    Relative to a fixed origin, O, the line l has the equation

    r = (i + pj − 5k) + λ(3i − j + qk),

    where p and q are constants and λ is a scalar parameter.

    Given that the point A with coordinates (−5, 9, −9) lies on l,

    (a) find the values of p and q,

    (b) show that the point B with coordinates (25, −1, 11) also lies on l.

    The point C lies on l and is such that OC is perpendicular to l.

    (c) Find the coordinates of C.

    (d) Find the ratio AC : CB

    I have no problem with parts a to c (for a the answer is p=7, q=2, and for c the answer is (4, 6, −3)). However, for part (d) the method which I used was simply finding the magnitude of AC and CB, and then finding the ratio. However, the markscheme has used a far more elegant method, which I was wondering if someone could explain:

    A : λ = −2, B : λ = 8, C : λ = 1 ∴ AC : CB = 3 : 7

    Thank you for any assistance.
    Last edited: Jun 8, 2008
  2. jcsd
  3. Jun 8, 2008 #2


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    I take it you mean that in parts (a), (b), and (c), you have determined that the corresponding [itex]\lambda[/itex] values for A, B, and C are -2, 8, and 1 respectively.

    Write the equation of the line as P+ tQ. Then A= P-2Q, B= P+ 8Q, C= P+ Q. A-C is the vector (-2-(1))Q= -3Q and B- C is the vector (8- 1)Q= 7Q. Since the lengths AC and BC are 3|Q| and 7|Q| respectively the ratio of their lengths is 3/7.
  4. Jun 8, 2008 #3
    ahh yes, of course! Thank you!
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