Linear algebra help dealing with the equation of a plane....please see attachment

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Linear algebra help dealing with the equation of a plane....please see attachment

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I need help with 1.12, I have tried the problem twice and no luck proving it
which part do u need help with the cartesian equation or the proof

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the first part of the question is simple getting you familiar with transforming a set of parametric equations into a cartesian equation
perhaps reading your notes on this will jog your memory
Bout to upload what I did give me just a second
@dan815 I Know if I can get both of these planes into Cartesian equations I can show that the normal vector is a scalar multiple of the other and that will prove the planes are parallel. Then showing any point on both planes is the same will prove they are the same plane. The problem is I can't get the Cartesian equations in a way that shows the normal vectors are the same. If the normal vectors are not the same then they can't be the same plane
@chris00 thanks for the tip about reading my notes, that never occurred to me
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ah okay thats quite a bit of work
Is there an easier way?
the point 1,1,1 should satisfy both equations correct?
Well yes, but it doesn't prove they are the same plane
we will use that fact to prove it
Ok
all we need are 2 things to be satisfied, both of them should have the same normal direction, and one point that is common,
agreed
from the 2 equations you derived, i can say there has been a mistake as both of the normal lines are not equal
That is what I thought but i can't see where. I actually did it twice and got the same exact thing so I figured there must be a mistake in the logic I used to derive the equations rather than a mistake in arithmetic or algebra
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Because I came up with the same two equations both times
okay can u compute that, and show mw the 2 normal vectors you get again
The way you annotated it?
I get what your saying, and it makes perfect sense, but this professor wants us to show the Cartesian equations and then use the normal vectors in those equations to show they are parallel normal vectors.
And this problem follows along those lines, because the first step is to find the Cartesian equation of the first line.
yep u can get the cartesian equations this way too
suppose you have a normal vector then your plane equations is ax+by+cz=d now you can solve for d, by plugging in one of your known points for example the (1,1,1)
you repeat the same process for equation 2
Ah, I see where your going and I didn't think of it that way. Let me try that way and see if I come up with an answer. Thank you
then you will see that a point in equation 1 exists in equation 2 as well, then these 2 places must coincide, because any 2 planes with the same normal vector direction will be parralel and can* only have a similiar point if they are in fact coincident
so you can check your answers, the first parametric eq has carteisian form: \[3x _{1}-4x _{2}+2_{3}=1\] second one has cartesian form: \[-24x _{1}+32x _{2}-16x _{3}=-8\] now if u divide the second equation by -8 on both sides, you will see it is simply the cartesian equation of the first plane
but as @dan815 said, its important to find the normal vector in these questions
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