System of equations can be interpreted as intersection of 3 planes in 3 dimensional space

Sam007

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Hi Everyone,

I struggling with this question. What would be the answer and why is it so? Why are the other ones wrong?

Thank you for your time!
 

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tkhunny

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How do you know if planes are parallel? (Proportional Normal Vector?)
 

JeffM

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You can use brute force and solve the system of simultaneous equations by using basic algebra.

\(\displaystyle x_1 + 4x_2 + 6x_3 = 18.\)

\(\displaystyle x_1 - 2x_2 + x_3 = -\ 6.\)

\(\displaystyle 2x_1 + 14x_2 + 17x_3 = -\ 6.\)

\(\displaystyle x_1 = 18 - 4x_2 - 6x_3.\)

\(\displaystyle 18 - 4x_2 - 6x_3- 2x_2 + x_3 = -\ 6 \implies = 6x_2 + 5x_3 = 24.\)

\(\displaystyle 2(18 - 4x_2 - 6x_3) + 14x_2 + 17x_3 = -\ 6 \implies 36 + 6x_2 + 5x_3 = -\ 6 \implies\)

\(\displaystyle 6x_2 + 5x_3 = -\ 24.\)

\(\displaystyle 24 = 6x_2 + 5x_3 = -\ 24 \implies 24 = -\ 24.\)

Hmm. That seems implausible. What do you conclude? Why?
 

pka

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Hi Everyone,

I struggling with this question. What would be the answer and why is it so? Why are the other ones wrong?

Thank you for your time!
\(\displaystyle \left| {\begin{array}{*{20}{r}}
1&4&6\\
1&{ - 2}&1\\
2&{14}&{17}
\end{array}} \right| = 0\) SEE HERE
What does that tell you?
 

Sam007

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How do you know if planes are parallel? (Proportional Normal Vector?)
Two planes are said to be parallel when their normal vectors are parallel.

How do you know when normal vectors are parallel?
 

Sam007

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You can use brute force and solve the system of simultaneous equations by using basic algebra.

\(\displaystyle x_1 + 4x_2 + 6x_3 = 18.\)

\(\displaystyle x_1 - 2x_2 + x_3 = -\ 6.\)

\(\displaystyle 2x_1 + 14x_2 + 17x_3 = -\ 6.\)

\(\displaystyle x_1 = 18 - 4x_2 - 6x_3.\)

\(\displaystyle 18 - 4x_2 - 6x_3- 2x_2 + x_3 = -\ 6 \implies = 6x_2 + 5x_3 = 24.\)

\(\displaystyle 2(18 - 4x_2 - 6x_3) + 14x_2 + 17x_3 = -\ 6 \implies 36 + 6x_2 + 5x_3 = -\ 6 \implies\)

\(\displaystyle 6x_2 + 5x_3 = -\ 24.\)

\(\displaystyle 24 = 6x_2 + 5x_3 = -\ 24 \implies 24 = -\ 24.\)

Hmm. That seems implausible. What do you conclude? Why?
Does that mean there are no solutions for x?
 

Sam007

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\(\displaystyle \left| {\begin{array}{*{20}{r}}
1&4&6\\
1&{ - 2}&1\\
2&{14}&{17}
\end{array}} \right| = 0\) SEE HERE
What does that tell you?
Reading a bit online. Does that mean that all the lines are planes are parrallel?
 

Dr.Peterson

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There are two separate issues you have to consider: (1) how many solutions are there (you've seen there are none), and (2) why? There can be no solutions because all planes are parallel, or because one pair are parallel, or because the intersections of pairs of planes are parallel, though none of the planes are. So, how many are parallel?

What have you learned about determining whether two planes are parallel, given their equations? You mentioned their normal vectors. How are normal vectors related to the coefficients of the equations? And what does parallel mean for vectors?
 

Sam007

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There are two separate issues you have to consider: (1) how many solutions are there (you've seen there are none), and (2) why? There can be no solutions because all planes are parallel, or because one pair are parallel, or because the intersections of pairs of planes are parallel, though none of the planes are. So, how many are parallel?

What have you learned about determining whether two planes are parallel, given their equations? You mentioned their normal vectors. How are normal vectors related to the coefficients of the equations? And what does parallel mean for vectors?
Wow, you are so great at making things clear.

So in basic geometry, when lines are parallel they never intersect i.e have solutions.

Considering, there is no solution for x?

It means that all the planes are parallel

So the answer would be 3/C?
 

Dr.Peterson

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No, I explicitly told you that there are other ways for three planes not to have a single intersection point. See, for example, the pictures here.

The fact that if three planes are parallel, there is no solution is not the same as saying that if there is no solution, the three planes are parallel!

Read and answer all of my questions.
 

Sam007

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No, I explicitly told you that there are other ways for three planes not to have a single intersection point. See, for example, the pictures here.

The fact that if three planes are parallel, there is no solution is not the same as saying that if there is no solution, the three planes are parallel!

Read and answer all of my questions.
The visual makes it a bit more clear.

'What have you learned about determining whether two planes are parallel, given their equations? '- There is no solution to the equations

'How are normal vectors related to the coefficients of the equations? '- Not sure about that

'what does parallel mean for vectors?' one has to be the scalar multiple of the other

So we know there are no solutions for x.

That means, that it could be

1. Three parallel planes
2. two parallel planes and one intersecting plane
3. three planes that intersect the other two but not at the same location

So I have narrowed it down to 3,5,6?

How do I which one it is though?
 

Dr.Peterson

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'What have you learned about determining whether two planes are parallel, given their equations? '- There is no solution to the equations
No; the question was, from the equations of two planes, how do you find whether they are parallel? You don't have to solve anything.

This has already been mentioned: compare their normal vectors. You must have learned about this, or you would not be given this question. Do you know what normal vectors are?

If you don't, then I suppose you could try solving each pair of equations together, and see what happens. But that's a lot harder.

'How are normal vectors related to the coefficients of the equations? '- Not sure about that
Look here. If you don't follow it, then please tell us what you have learned, so we will have some idea what to expect of you.

This should have been mentioned before now: read our submission guidelines! This might not take so long if you had introduced yourself more fully, including telling us what you know about the subject.

'what does parallel mean for vectors?' one has to be the scalar multiple of the other
Correct. So once you find the normal vectors, you should be in business.
 

pka

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So in basic geometry, when lines are parallel they never intersect i.e have solutions.
Considering, there is no solution for x?
It means that all the planes are parallel
So the answer would be 3/C?
Here we have three planes:
\(\displaystyle \begin{align*}\text{ i. }x+4y+6z&=18 \\\text{ ii. }x-2y+z&=-6\\\text{iii. }2x+4y+17z&=-6 \end{align*}\)
Because no two of those planes are parallel then each pair have a line of intersection.
Planes plane i. has normal \(\displaystyle N_1=<1.4.6>\) likewise \(\displaystyle N_2=<1,-2,1>\text{ & }N_3=<2,14,17>\).
Then \(\displaystyle N_1\times N_2=<16,-5,-6\) is the direction vector for the the intersection of planes \(\displaystyle \text{ i. & ii. }\).
Likewise, \(\displaystyle N_1\times N_3=<-16,5,6>\) is the direction vector for the the intersection of planes \(\displaystyle \text{ i. & iii. }\).
Because those two lines are parallel and we know that there is no point common to all three planes what is your answer.?
 

Sam007

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Here we have three planes:
\(\displaystyle \begin{align*}\text{ i. }x+4y+6z&=18 \\\text{ ii. }x-2y+z&=-6\\\text{iii. }2x+4y+17z&=-6 \end{align*}\)
Because no two of those planes are parallel then each pair have a line of intersection.
Planes plane i. has normal \(\displaystyle N_1=<1.4.6>\) likewise \(\displaystyle N_2=<1,-2,1>\text{ & }N_3=<2,14,17>\).
Then \(\displaystyle N_1\times N_2=<16,-5,-6\) is the direction vector for the the intersection of planes \(\displaystyle \text{ i. & ii. }\).
Likewise, \(\displaystyle N_1\times N_3=<-16,5,6>\) is the direction vector for the the intersection of planes \(\displaystyle \text{ i. & iii. }\).
Because those two lines are parallel and we know that there is no point common to all three planes what is your answer.?
My peer had this alternative solution, what are your thoughts
 

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pka

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My peer had this alternative solution, what are your thoughts
"What are my thoughts?" Did you understand my post?
I agree with your friend. I said that no two of the planes are parallel.
So any two must intersect, but there is no point common to all three.
Therefore, option #5 is the only one that can be correct.
 
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