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Vectors (Intersecting Lines)

  1. Apr 6, 2013 #1


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    1. The problem statement, all variables and given/known data

    This is taken from STEP II 2003 Q5.

    The position vectors of the points A, B and P with respect to an origin O are ai , bj and li + mj + nk , respectively, where a, b, and n are all non-zero. The points E, F, G and H are the midpoints of OA, BP, OB and AP, respectively. Show that the lines EF and GH intersect.

    Let D be the point with position vector dk, where d is non-zero, and let S be the point of intersection of EF and GH. The point T is such that the mid-point of DT is S. Find the position vector of T and hence find d in terms of n if T lies in the plane OAB.

    2. Relevant equations


    3. The attempt at a solution

    I can't seem to make any headway with this at all, some guidance would be appreciated. There's a solution given which I don't understand:

    I agree with E = a/2.

    I don't understand how F = (p+b)/2. If OB + BP = OP, then BP = OP - OB, so F = ½BP = ½(p-b), not ½(p+b)?

    I agree with G = b/2.

    I don't understand how H = (p+a)/2, for the same reason that I don't agree with F = (p+b)/2. Can anyone help?

    I must be doing something very wrong because I am getting that EF and GH are identical (so intersect everywhere).
    Last edited: Apr 6, 2013
  2. jcsd
  3. Apr 6, 2013 #2


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    What does that mean? In the statement of the problem, above, you use bold face to mean vectors but the only "a" given is a number, not a vector.

  4. Apr 6, 2013 #3


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    Apologies, I didn't make it clear -- a in non-bold text is just the constant a, whereas a or p indicate the vectors OA or OP.
  5. Apr 7, 2013 #4


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    Look at an example. If B= (1, 0, 0) and P= (0, 1, 0) the the point half way between them is the "average" ((1+0)/2, (0+1)/2, (0+0)/2)= (1/2, 1/2, 0). The vector form would be <1/2, 1/2, 0>= (1/2)<1, 0, 0>+ (1/2)<0, 1, 0>.

    The difficulty with your reasoning is that "(1/2)BP" is NOT the vector from O to that midpoint. Yes, we can think of BP as a vector from B to P and (1/2)BP as a vector from B to the midpoint of BP. To represent (1/2)BP as a vector from O, we have to add B: <1, 0, 0>+ <-1/2, 1/2, 0>= <1/2, 1/2, 0>.
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