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

[Mtl]

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.

 

[HallsofIvy]

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?

 

[Mtl]

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

Yes, the negative affects the direction, or so my textbook says.

Anyways the first answer would look something along the lines of
X__T____Y__Z

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