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

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In summary, the first part asks for a diagram which satifies the following X divides YZ in a ration of -5:7 and also that T divides XY in the ratio 1:3. The next part states that if O is not on the line, write OT as a linear combination of OX and OY. The last part asks for OY as a linear combination of OT and OZ. 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
Mtl
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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.
 
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  • #2
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?
 
  • #3
"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.
 

1. What does it mean for X to divide YZ and T to divide XY?

When we say that X divides YZ and T divides XY, it means that both X and T are factors of YZ and XY respectively. In other words, X and T are numbers that, when multiplied by another number, result in YZ and XY.

2. What is a linear combination?

A linear combination is a mathematical expression that consists of a sum of terms, each of which is a constant multiplied by a variable. In other words, it is a way of expressing a number as a combination of other numbers, using addition and multiplication.

3. How can we write OT as a linear combination of OX and OY?

To write OT as a linear combination of OX and OY, we need to find two constants, let's say a and b, such that a*OX + b*OY = OT. These constants can be found by solving the equations X*a + Y*b = T and Y*a + Z*b = 0, where X, Y, and Z are the factors of YZ and T that we mentioned earlier.

4. Why is it important to write OT as a linear combination of OX and OY?

Writing OT as a linear combination of OX and OY allows us to represent a complex equation in a simpler form. It also helps us to better understand the relationship between the different variables and how they are related to each other.

5. Can we write any number as a linear combination of other numbers?

Yes, any number can be written as a linear combination of other numbers. This is because a linear combination is simply a way of representing a number using addition and multiplication, which are fundamental operations in mathematics.

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