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

In summary, the conversation discusses the positions of three towns (A, B, and X) and the path of an airplane from town A to B. The question is to find the distance between points X and Y, with Y being the closest point on the path AB to town X. The method for finding this distance is explained, but there is a mistake in the calculations resulting in a discrepancy with the book's answer.
  • #1
Peter G.
442
0
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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  • #2
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!
 
  • #3
Sorry, I meant:

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

The calculations were performed based on 180 not 280t
 
  • #4
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?
 
  • #5
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
 

1. What are vectors and how are they used in science?

Vectors are mathematical quantities that have both magnitude and direction. They are used in many scientific fields, including physics, engineering, and biology, to represent physical quantities such as velocity, force, and displacement.

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