# Math Help - application of the distance formula in 3d

1. ## application of the distance formula in 3d

Hello, I'm stuck in an essay that I am writing about the Pythagorean theorem and the Distance Formula.

I am directed to describe real world applications for the use of the distance formula in 2 and 3 d. I'm good on 2d, but struggling with 3 d.

I've gotten lots of ideas that use just the P theorem, (what i mean, is where you already know the length of two sides and want to find the diaganol, for example)... but none that fit the assignment which is:

"The distance formula needs to be applied to at least two real-life examples, which means coordinate axes need to be defined and coordinates need to be assigned to the points." (in 3d)

I appreciate any direction that can be provided.

2. Originally Posted by YouCanCallMeBianca
Hello, I'm stuck in an essay that I am writing about the Pythagorean theorem and the Distance Formula.

I am directed to describe real world applications for the use of the distance formula in 2 and 3 d. I'm good on 2d, but struggling with 3 d.

I've gotten lots of ideas that use just the P theorem, (what i mean, is where you already know the length of two sides and want to find the diaganol, for example)... but none that fit the assignment which is:

"The distance formula needs to be applied to at least two real-life examples, which means coordinate axes need to be defined and coordinates need to be assigned to the points." (in 3d)

I appreciate any direction that can be provided.
What about trying to fit the longest possible rod into a rectangular truck?

3. thanks, but I don't see how I can put that onto a coordinate system. In the case of a truck, it's dimensions are already known, i don't need to use the distance formula, I can just use the length and width to find the diagonal.

4. Originally Posted by YouCanCallMeBianca
Hello, I'm stuck in an essay that I am writing about the Pythagorean theorem and the Distance Formula.

I am directed to describe real world applications for the use of the distance formula in 2 and 3 d. I'm good on 2d, but struggling with 3 d.

I've gotten lots of ideas that use just the P theorem, (what i mean, is where you already know the length of two sides and want to find the diaganol, for example)... but none that fit the assignment which is:

"The distance formula needs to be applied to at least two real-life examples, which means coordinate axes need to be defined and coordinates need to be assigned to the points." (in 3d)

I appreciate any direction that can be provided.
Suppose an aircraft is coming in to land in a fierce cross-wind, and the autopilot is engaged to land the aircraft. The autopilot needs to know what attitude it needs to sit it to get the correct vertical and forward velocity. And in order to do this, it needs to know the 3D distance from the current position to the base of the runway.

The aircraft simply treats it's the position of the runway base as the origin, obtained from radar or other communication, and gets it's own coordinate from there via GPS.

5. Another example could be calculating the shortest distance from the launch pad to the edge of the atmosphere for a spacecraft in order to reduce the fuel needed in the rocket stages.

6. Originally Posted by Mush
Another example could be calculating the shortest distance from the launch pad to the edge of the atmosphere for a spacecraft in order to reduce the fuel needed in the rocket stages.
ok, mush.

How do I define the coordinate system?

I think this is where I'm really getting stuck, is the coordinate system and being able to make sure I'm using units that relate to one another. And the more I think about it the more twisted up I'm getting.

(p/s - and wouldn't the answer be... straight up?)

7. Originally Posted by YouCanCallMeBianca
ok, mush.

How do I define the coordinate system?

I think this is where I'm really getting stuck, is the coordinate system and being able to make sure I'm using units that relate to one another. And the more I think about it the more twisted up I'm getting.

(p/s - and wouldn't the answer be... straight up?)
I suppose the answer would be 'straight up', yes.

What do you mean by defining the coordinate system? The coordinate system is already defined as 3D euclidian space, is it not? All you have to do is place the origin somewhere, and that's fairly arbitrary.

The example about the largest pole in a rectangular truck is a good one by the way. You're correct that you already know the dimensions of the truck, but you know the dimensions of imaginary box that you insert into the distance equation.

What the poster was saying is that the largest pole is determined by the distance between one corner and the opposite corner. You could calculate this using Pythagorus twice, OR you could calculate it using the distance formula once.

8. Originally Posted by Mush
I suppose the answer would be 'straight up', yes.

What do you mean by defining the coordinate system? The coordinate system is already defined as 3D euclidian space, is it not? All you have to do is place the origin somewhere, and that's fairly arbitrary.

The example about the largest pole in a rectangular truck is a good one by the way. You're correct that you already know the dimensions of the truck, but you know the dimensions of imaginary box that you insert into the distance equation.

What the poster was saying is that the largest pole is determined by the distance between one corner and the opposite corner. You could calculate this using Pythagorus twice, OR you could calculate it using the distance formula once.
Exactly.

Incidentally, suppose the truck has dimensions $l \times w \times h$.

Then its co-ordinates would be

$0 \leq x \leq l$

$0 \leq y \leq w$

$0 \leq z \leq h$...