Question

Find equation of the plane through (-1,4,2) and containing the line of intersection of the planes
4x-y+z-2=0 , 2x+y-2z-3=0

Answer 1

First do you know how to find the line of intersection of the 2 planes?

Answer 2

Yes!

Answer 3

So what is it?

Answer 4

we find the cross product of the normals of the two planes...and thats the vector parallel to normal of the two planes...then we find a point which lies on both the planes...???

Answer 5

Hmm that's not how I learned it, you may be correct though. Actually yes that makes sense. This is how I would do it. Simply solve the matrix that is the solution set for both planes

4x-y+z=2
2x+y-2z=3

[4 -1 1 2]
[2 1 -2 3]

Row reduce that and you get

[1 0 -1/6 5/6]
[0 1 -5/3 4/3]

The solution of this is:

(5/6, 4/3, 0) + t*(1/6, 5/3, 1)
=(5/6, 4/3, 0) + t*(1, 10, 6)

You can try it your way and you should get the same thing

Answer 6

OK now you have a vector on the plane (1,10,6) and you have 2 points: (-1,4,2) and (5/6, 4/3, 0). Any idea what to do next?

Answer 7

okay i got till here...after that i just couldnt figure out what to do :/

Answer 8

Why is your picture hitler??? Kind of offensive.

Anyways, if you find 2 vectors that are on a plane, you can always find the equation for the plane by simply calculating their cross product. You have 1 vector, and you can get a second vector by simply finding the difference between the 2 points!

Answer 9

Don't peak if you want to try it on your own! Here's my work:



(-1,4,2) - (5/6, 4/3, 0) = (-11/6,8/3,2)

(1,10,6) x (-11/6,8/3,2) = (4,-13,21)

4x-13y+21z = d
4(-1)-13(4)+21(2) = d
-14 = d

4x-13y+21z = -14
:)

Answer 10

Okay I have another question. If the plane was perpendicular to the line of intersection of the two planes... what would we do then? will the answer be the same as this one?

Answer 11

How can a plane be perpendicular to a line?

Answer 12

I don't know :/
I recently saw a question like that but can't find it now....

Answer 13

A plane can't be perpendicular to a line, that makes no sense...

Answer 14

oh i meant to say containing the line perpendicular to line of intersection of two planes

Answer 15

sorry my bad

Answer 16

That's fine! I just took linear last semester and I was SO confused. A plane can be perpendicular to a plane.

Answer 17

You seem to be doing very well, this geometry stuff throws most people off but you're on track :)

Answer 18

Oh you corrected yourself. I misread. "containing the line perpendicular to line of intersection of two planes". Hmm gimme a min

Answer 19

Because there are infinite lines that are perpendicular to a line in 3 dimensions... I'm not sure sorry

Answer 20

Okayy. Np. Thanks for your help. :)