# Vectors and finding points on a line

Long John Silver, a pirate, has buried his treasure on an island with five trees located at the following points: A (30.0m, -20.0m), B(60.0 m, 80.0m), C (-10.0m, -10.0m), D (40.0m, -30.0m), and E(-70.0m, 60.0m). All of the points are measured relative to some origin. Long John's map instructs you to start at A and move toward B, but cover only one-half the distance between A and B. Then move toward C, covering one-third the distance between your current location and C. Then move toward D, covering one-fourth the distance between where you are and D. Finally, move toward E, covering one-fifth the distance between you and E, stop and dig. (a) What are the coordinates of the point where the pirate's treasure is buried?...after trying several formulas that I know and looking at the internet for help, I haven't been able to find an answer. If anyone can please help me it will be much appreciated.

-
yes, but I don't know how that will help me find the points – Ronnie.j Sep 11 '11 at 22:10
@mixedmath: You don't need the distance formula here; you can just take the given fractions of the vectors without calculating their lengths. – joriki Sep 11 '11 at 22:22
@joriki: oh, you're absolutely right. I was busy thinking that this is just some computation, and I gave the wrong one. Very silly of me. Thank you. – mixedmath Sep 11 '11 at 22:30

Hint: The $x$ coordinate of the point a fraction $p$ from $S$ to $T$ is $x_S +p(x_T-x_S)$, and similarly with the $y$ coordinate. Just work through the calculations

Here is a diagram to help (I think)

-
THanks, i understand the concept of the method but I have difficulty in trying to explain how this method works. Can you help me? – Ronnie.j Sep 11 '11 at 22:31
@Ronnie: Can you explain what the coordinates of $P_2$ are and why? – Henry Sep 11 '11 at 22:36
Following the hint, A is (30,-20) and B is (60,80). So B-A is (30,100). Half way from A to B is (30,-20) + (30,100)/2=(45,30), shown as P2. – Ross Millikan Sep 11 '11 at 22:38
Yes i know that, i got my answer using his method. Im just stuck as to explaining why his method works – Ronnie.j Sep 11 '11 at 22:39
@Ronnie: it really is no more complicated than "starting at $A$ and going half of the way to $B$ from $A$". – Henry Sep 11 '11 at 22:48

The locations are:

P1 = A + (B - A) / 2
P2 = P1 + (C - P1) / 3
P3 = P2 + (D - P2) / 4
P4 = P3 + (E - P3) / 5    == treasure location


Each equation are in fact 2 equations for the x and y coordinates respectively.

B - A is the vector from A to B. When you are in A and move along the vector B - A you arrive at A + (B - A) = B. When you move only half of the distance you arrive at P1 = A + (B - A)/2.

-
thanks alot, it works very well. If you were to explain this though, how would i explain it? i know the concepts of it, but i wouldnt be able to explain it – Ronnie.j Sep 11 '11 at 22:25
I added a simple explanation. You might look at some vector geometry in the plane (2D), eg. Wikipedia. – Jiri Sep 11 '11 at 22:36
thank you, this makes alot of sense now – Ronnie.j Sep 11 '11 at 22:45
another question, how did you get B-A? like i know why you would make it B-A, but if someone asks you, why did you call the vector B-A, what would you say? – Ronnie.j Sep 11 '11 at 22:52
or is it something you just have to know? – Ronnie.j Sep 11 '11 at 22:56