What is the equation for the line AB and the distance XY?

AI Thread Summary
The discussion centers on determining the equation for the line AB between towns A and B, and calculating the distance from point Y on this line to town X. Town A is located 240 km East and 70 km North of origin O, while town B is 480 km East and 250 km North. The user initially miscalculated the direction vector for line AB but later corrected it to reflect the proper coordinates. After finding point Y's coordinates, the user calculated the distance XY as 19 km, which contradicts the book's claim of 75 km. The user ultimately resolved that their method was correct, but the initial numerical errors led to the discrepancy.
Peter G.
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Hi,

So, there are three towns: A, B and X

Town A is 240 km East and 70 km North of O.
Town B is 480 km East and 250 km North of O.
Town X is 339 km East and 238 km North of O.

At A, the airplane changes direction so it now flies towards B. Point Y in the path AB is the closest the airplane ever is to town X.

They then ask us to show that AB is perpendicular to -3i + 4j, which I can do.

They then ask us to find Distance XY.

To do so, I did the following:

r = (240 + 240t) + (140+280t)

r = (339 - 3s) + (238 + 4s)

I then solved to find the point of intersection, which should, supposedly, yield the position of Y.

I got t as 0.46, meaning Y would be 350.4i + 222.8j.

I then went on to find the vector XY and work out its magnitude, which gave me 19 km.

The book, however, claims it is 75km.

Can anyone help me spot where I went wrong?

Thanks in advance!
 
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Peter G. said:
Hi,

So, there are three towns: A, B and X

Town A is 240 km East and 70 km North of O.
Town B is 480 km East and 250 km North of O.
Town X is 339 km East and 238 km North of O.

At A, the airplane changes direction so it now flies towards B. Point Y in the path AB is the closest the airplane ever is to town X.

They then ask us to show that AB is perpendicular to -3i + 4j, which I can do.

They then ask us to find Distance XY.

To do so, I did the following:

r = (240 + 240t) + (140+280t)
What is your reasoning for the above?
Peter G. said:
r = (339 - 3s) + (238 + 4s)

I then solved to find the point of intersection, which should, supposedly, yield the position of Y.

I got t as 0.46, meaning Y would be 350.4i + 222.8j.

I then went on to find the vector XY and work out its magnitude, which gave me 19 km.

The book, however, claims it is 75km.

Can anyone help me spot where I went wrong?

Thanks in advance!
 
Sorry, I meant:

r= (240+240t) + (140+180t).

The calculations were performed based on 180 not 280t
 
Peter G. said:
Sorry, I meant:

r= (240+240t) + (140+180t).

The calculations were performed based on 180 not 280t
What I'm asking is, where does this equation come from, especially the 240t and 180t terms?
 
Sorry, I got my mistake now. The method was right, I just had gotten the numbers wrong again.

I'll explain that equation (which should read (240 + 240t)+(70+180t)

It is the equation for the line AB. A point on the line (A) is 240i + 70j whereas the direction, AB, is 240i + 180j
 

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