1. ## rectangular coordinate system

I don't understand this problem but i am still working on it....

2. ## Re: rectangular coordinate system

Hi Veronica!

It seems to me that the right answer is missing...

But to find out, suppose the line is of the form $\displaystyle \mathbf{a} + \lambda \mathbf{b}$ with $\displaystyle \lambda > 0$.
What are the options for $\displaystyle a_x$?
And what are the corresponding options for $\displaystyle b_x$?

3. ## Re: rectangular coordinate system

Originally Posted by ILikeSerena
It seems to me that the right answer is missing...
But to find out, suppose the line is of the form $\displaystyle \mathbf{a} + \lambda \mathbf{b}$ with $\displaystyle \lambda > 0$.
What are the options for $\displaystyle a_x$?
And what are the corresponding options for $\displaystyle b_x$?
I agree that the right answer is not there,
But what is the right question?
As is, its wording is extremely odd.
It seems to me the question is about line segments made of exactly four points taken from a $\displaystyle 3D$ lattice where the points are the 64 latice points, from $\displaystyle (1,1,1)\text{ to }(4,4,4)$.
We have three sets of four planes with ten lines in each plane.
BUT I don't know how we should count the diagonals.
Any thoughts?

4. ## Re: rectangular coordinate system

Originally Posted by Plato
I agree that the right answer is not there,
But what is the right question?
As is, its wording is extremely odd.
It seems to me the question is about line segments made of exactly four points taken from a $\displaystyle 3D$ lattice where the points are the 64 latice points, from $\displaystyle (1,1,1)\text{ to }(4,4,4)$.
We have three sets of four planes with ten lines in each plane.
BUT I don't know how we should count the diagonals.
Any thoughts?
Umm... I don't see any ambiguity in the question.
It asks for lines and not for line segments.
I interpret it to ask for all possible lines with 4 distinct points on it with integer coordinates between 1 and up to 4 (your 64 lattice points).

5. ## Re: rectangular coordinate system

Originally Posted by ILikeSerena
Umm... I don't see any ambiguity in the question.
It asks for lines and not for line segments.
I interpret it to ask for all possible lines with 4 distinct points on it with integer coordinates between 1 and up to 4 (your 64 lattice points).
But that is exactly the same set of lines in either case.
What about diagonals? Are there more than four?

6. ## Re: rectangular coordinate system

All I know is the answer is (D) 76.
And I am still trying to understand the problem...

7. ## Re: rectangular coordinate system

Originally Posted by Plato
But that is exactly the same set of lines in either case.
What about diagonals? Are there more than four?
I don't understand what puzzles you...
In my interpretation there are 4 3D-diagonals, and a number more of 2D-diagonals.

Originally Posted by Veronica1999
All I know is the answer is (D) 76.
And I am still trying to understand the problem...
In my count I'm not getting 76 lines.
Actually, I'm getting 61 lines.
(And you didn't answer my question.)

Edit: Hold on, I made a mistake counting...
Yep, I'm getting 76 lines now.

8. ## Re: rectangular coordinate system

Plato had a good point, counting the diagonals.
How many 3D diagonals are there?
How many 2D diagonals?
And how many other lines that have 4 distinct points on them with integer coordinates between 1 and up to 4?

9. ## Re: rectangular coordinate system

Thanks to everyone's help, I now understand the problem.
But i seem to get 73 lines.

10. ## Re: rectangular coordinate system

Hmm... how are you counting?
For starters, how many lines do you get parallel to the x-axis?
Did you count the lines that are going through the middle of the cube (and not through any corner or edge)?

11. ## Re: rectangular coordinate system

Originally Posted by ILikeSerena
Hmm... how are you counting?
For starters, how many lines do you get parallel to the x-axis?
Did you count the lines that are going through the middle of the cube (and not through any corner or edge)?
Let me set forth how I see it. Consider a $\displaystyle 2D\text{ 4-lattice}$.

We can consider that as a plane parallel to the $\displaystyle yz$-plane. All coordinates look like $\displaystyle (1,j,k),~1\le j,k\le 4$.
There are ten of our lines in that plane: four parallel to the $\displaystyle y$-axis; four parallel to the $\displaystyle z$-axis; and two diagonals.
There are three copies of that plane, $\displaystyle (x,j,k),~1\le j,k\le 4~\&~x=2,~3,~4$, all parallel to the $\displaystyle yz$-plane.
Now there are forty of our lines.

There are two more sets of those planes: 4 parallel to the $\displaystyle xz$-plane and 4 parallel to the $\displaystyle xy$-plane.

Now we are up to 120. We have 4 diagonals.

12. ## Re: rectangular coordinate system

Originally Posted by Plato
Let me set forth how I see it. Consider a $\displaystyle 2D\text{ 4-lattice}$.

We can consider that as a plane parallel to the $\displaystyle yz$-plane. All coordinates look like $\displaystyle (1,j,k),~1\le j,k\le 4$.
There are ten of our lines in that plane: four parallel to the $\displaystyle y$-axis; four parallel to the $\displaystyle z$-axis; and two diagonals.
There are three copies of that plane, $\displaystyle (x,j,k),~1\le j,k\le 4~\&~x=2,~3,~4$, all parallel to the $\displaystyle yz$-plane.
Now there are forty of our lines.

There are two more sets of those planes: 4 parallel to the $\displaystyle xz$-plane and 4 parallel to the $\displaystyle xy$-plane.

Now we are up to 120. We have 4 diagonals.
You are counting lines more than once.
If we only look at lines parallel to the x-axis, there are 16.
Times 3 is 48 (and not 96).

13. ## Re: rectangular coordinate system

I finally got 72.
But is there a more logical approach?
And less time consuming?
Where am I missing 4 lines?

14. ## Re: rectangular coordinate system

I got 76. Is my answer correct now?

15. ## Re: rectangular coordinate system

Hmm, it seems you counted 5 lines everywhere, where you should have counted 4.
It seems you included coordinate zero, but your problem statement says "positive" integers.

So for instance the diagonals you drew are actually 3x(4+4)=24.
Parallel to the axis you have 4x4 lines that intersect a face of your cube, which gives 3x4x4=64 lines.
Then your final 4 3D-lines gives 24+64+4=76.

Here's another way.
Suppose you represent each line by $\displaystyle \mathbf{a} + \lambda \mathbf{b}$ with $\displaystyle \lambda \ge 0$.
Then $\displaystyle \lambda$ must take the values 0,1,2,3 to make sure all coordinates are integers.

Then $\displaystyle a_x$ is either 1,2,3, or 4.
And $\displaystyle b_x$ can be zero in all cases.
If $\displaystyle a_x=1$ then $\displaystyle b_x$ can also be 1.
And if $\displaystyle a_x=4$ then $\displaystyle b_x$ can also be -1.

So there are 6 possibilities.
The same for y and z gives you $\displaystyle 6^3$ possibilities.

But b cannot be the zero vector, because then you would not have distinct points.
So $\displaystyle 4^3$ possibilities are wrong.

This gives you $\displaystyle 6^3 - 4^3$ possibilities.
Since you count every line double, you need to divide this result by 2...