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If X divides YZ, T divides XY, write OT as linear comb. of OX & OY

  1. Jan 2, 2008 #1


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    Ok so I have an assignment for my Calculas/Geometry class and I seem to have not paid the greatest attention in class as I'm pretty much completely lost. I was wondering if anyone could help me with one of the questions.

    The first part asked for a diagram which satified the following X divides YZ in a ration of
    -5:7 and also that T divides XY in the ratio 1:3. (I'm pretty sure I have correctly completed this part)

    The next part states that if O is not on the line, write OT as a linear combination of OX and OY.

    Also it says to write OY as a linear combination of OT and OZ.
  2. jcsd
  3. Jan 3, 2008 #2


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    Is that really a ratio of negative 5 to 7? Since lengths are never negative, how did you show that? Assuming that is 5:7, you have a line segment, YZ, divided into 5+ 7= 12 parts and X is 5 of those parts from Y. To put T dividing XY in the ratio 1:3, you would need to divide the segment XY into 4 parts. To simplify things, divide the original YZ into 4(12)= 48 parts and mark X 4(5)= 20 parts from Y (and so 4(7)= 28 parts from Z: 20:28= 5:7). Now mark T 5 parts from X, so 20-5= 15 parts from Y: 5:15= 1:3.

    Now assume you have some "origin", O, not on that line. Think of OT as the vector from O to T, OX as the vector from O to X, and OY as the vector from O to Y. Now, you can write the various vectors as sums and difference of the others- follow along on your diagram. OT= OY+ YT and OT= OX- XT. Adding those equations, 2OT= OX+ OY+ YT- XT. Since T divides XY in the ratio 1:3, OX= (1/4)XY and OY= (3/4)XY so YT- XT= (3/4- 1/4)XY= (1/2)XY. That is, 2OT= OX+ OY+ (1/2)XY. XY= OX- OY, so (1/2)XY= (1/2)OX- (1/2)OY and we have 2OT= OX+ OY+ (1/2)(OX - OY)= (3/2)OX+ (1/2)OY. Dividing through by 2,OT= (3/4)OX+ (1/4)OY.

    Now, can you do "OY as a linear combination of of OT and OZ?
  4. Jan 3, 2008 #3


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    "Is that really a ratio of negative 5 to 7?"

    Yes, the negative affects the direction, or so my text book says.

    Anyways the first answer would look something along the lines of

    Although, through reading through what you have read I'm pretty sure i can finish the problem, so thank you for your help.
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