\(\displaystyle \left| {\begin{array}{*{20}{r}}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!
How do you know if planes are parallel? (Proportional Normal Vector?)
You can use brute force and solve the system of simultaneous equations by using basic algebra.
[MATH]x_1 + 4x_2 + 6x_3 = 18.[/MATH]
[MATH]x_1 - 2x_2 + x_3 = -\ 6.[/MATH]
[MATH]2x_1 + 14x_2 + 17x_3 = -\ 6.[/MATH]
[MATH]x_1 = 18 - 4x_2 - 6x_3.[/MATH]
[MATH]18 - 4x_2 - 6x_3- 2x_2 + x_3 = -\ 6 \implies = 6x_2 + 5x_3 = 24.[/MATH]
[MATH]2(18 - 4x_2 - 6x_3) + 14x_2 + 17x_3 = -\ 6 \implies 36 + 6x_2 + 5x_3 = -\ 6 \implies[/MATH]
[MATH]6x_2 + 5x_3 = -\ 24.[/MATH]
[MATH]24 = 6x_2 + 5x_3 = -\ 24 \implies 24 = -\ 24.[/MATH]
Hmm. That seems implausible. What do you conclude? Why?
Reading a bit online. Does that mean that all the lines are planes are parrallel?\(\displaystyle \left| {\begin{array}{*{20}{r}}
1&4&6\\
1&{ - 2}&1\\
2&{14}&{17}
\end{array}} \right| = 0\) SEE HERE
What does that tell you?
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?
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.
'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
Here we have three planes: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?
My peer had this alternative solution, what are your thoughtsHere 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.?
"What are my thoughts?" Did you understand my post?My peer had this alternative solution, what are your thoughts