Question

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

Answer 1

I need help with 1.12, I have tried the problem twice and no luck proving it

Answer 2

which part do u need help with the cartesian equation or the proof

Answer 3

the first part of the question is simple getting you familiar with transforming a set of parametric equations into a cartesian equation

Answer 4

perhaps reading your notes on this will jog your memory

Answer 5

Bout to upload what I did give me just a second

Answer 6

@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

Answer 7

@chris00 thanks for the tip about reading my notes, that never occurred to me

Answer 8

ah okay thats quite a bit of work

Answer 9

Is there an easier way?

Answer 10

the point 1,1,1 should satisfy both equations correct?

Answer 11

Well yes, but it doesn't prove they are the same plane

Answer 12

we will use that fact to prove it

Answer 13

Ok

Answer 14

all we need are 2 things to be satisfied, both of them should have the same normal direction, and one point that is common,

Answer 15

agreed

Answer 16

from the 2 equations you derived, i can say there has been a mistake as both of the normal lines are not equal

Answer 17

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

Answer 18

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Answer 19

Because I came up with the same two equations both times

Answer 20

okay can u compute that, and show mw the 2 normal vectors you get again

Answer 21

The way you annotated it?

Answer 22

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.

Answer 23

And this problem follows along those lines, because the first step is to find the Cartesian equation of the first line.

Answer 24

yep u can get the cartesian equations this way too

Answer 25

suppose you have a normal vector <a,b,c>
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)

Answer 26

you repeat the same process for equation 2

Answer 27

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

Answer 28

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

Answer 29

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

Answer 30

but as @dan815 said, its important to find the normal vector in these questions