# Thread: point lie in plane

1. ## point lie in plane

does the point $(4,5,6)$ lie in the plane $(x,y,z)=(4,1,6)+p(3,-2,1)+q(-6,6,-1)
$

i started by writing the parametric equations

$x=4+3p-6q$
$y=1-2p+6q$
$z=6+p-q$

subbing in $(4,5,6)$ into the corresponding equations

$4=4+3p-6q$
$5=1-2p+6q$
$6=6+p-q$

simplifying gives

$0=3p-6q$
$4=-2p+6q$
$0=p-q$ thus $p=q$

what do i do now? i know i need to figure out p and q i just cant figure it out

2. Originally Posted by william
does the point $(4,5,6)$ lie in the plane $(x,y,z)=(4,1,6)+p(3,-2,1)+q(-6,6,-1)
$

i started by writing the parametric equations

$x=4+3p-6q$
$y=1-2p+6q$
$z=6+p-q$

subbing in $(4,5,6)$ into the corresponding equations

$4=4+3p-6q$
$5=1-2p+6q$
$6=6+p-q$

simplifying gives

$0=3p-6q$
$4=-2p+6q$
$0=p-q$ thus $p=q$

what do i do now?
You need to check that $p$ can equal $q$.

What happens if you plug it into the other two equations?

3. Originally Posted by william
does the point $(4,5,6)$ lie in the plane $(x,y,z)=(4,1,6)+p(3,-2,1)+q(-6,6,-1)
$

i started by writing the parametric equations

$x=4+3p-6q$
$y=1-2p+6q$
$z=6+p-q$

subbing in $(4,5,6)$ into the corresponding equations

$4=4+3p-6q$
$5=1-2p+6q$
$6=6+p-q$

simplifying gives

$0=3p-6q$
$4=-2p+6q$
$0=p-q$ thus $p=q$

what do i do now? i know i need to figure out p and q i just cant figure it out
i really don't get how to check

for example if i plug into the first equation

0=3p-6p
0=3p
0=p

4. Originally Posted by william
i really don't get how to check

for example if i plug into the first equation

0=3p-6p
0=3p
0=p
Okay so now you know that $p=q=0$ . This says the the point $(4,5,6)=...$ is this true?

5. Originally Posted by TheEmptySet
Okay so now you know that $p=q=0$ . This says the the point $(4,5,6)=...$ is this true?
its just the answers in my book show p=4 and q=2 im confused how theyre getting that because i'm simply not getting anything close to that

6. Originally Posted by william
its just the answers in my book show p=4 and q=2 im confused how theyre getting that because i'm simply not getting anything close to that
Plug p and q into what you started with

$(x,y,z)=(4,1,6)+p(3,-2,1)+q(-6,6,-1)$

$(x,y,z)=(4,1,6)+4(3,-2,1)+2(-6,6,-1)$

$(x,y,z)=(4,1,6)+(12,-8,4)+(-12,12,-2)=(4,5,8)$

So solution does not check. So something is wrong either in the book or in the problem wne it was written down.

7. Originally Posted by TheEmptySet
Plug p and q into what you started with

$(x,y,z)=(4,1,6)+p(3,-2,1)+q(-6,6,-1)$

$(x,y,z)=(4,1,6)+4(3,-2,1)+2(-6,6,-1)$

$(x,y,z)=(4,1,6)+(12,-8,4)+(-12,12,-2)=(4,5,8)$

So solution does not check. So something is wrong either in the book or in the problem wne it was written down.

the book says that p=4 and q=2, as answers. these i am assuming are supposed to be obtained by substitution into the parametric equations, as we were doing earlier, but i do not know how to obtain those numbers. also, since they do not match up, does that not simply mean that the point does not lie in the plane?

8. if you were wondering, i did solve the equations to find p and q, i finally figured out. i simply subtracted two of the equations using the elimination method.