Three Planes Defined By 3 Equations (P6 Stuff)

I don't understand what 3 planes defined by 3 equations are supposed to represent geometrically.

Is this right - If there is a unique solution to the 3 equations then this means that the 3 planes intersect a single point in common.

If there is no unique solution i.e.) The determinant of the matrix that corresponds to the set of 3 equations = 0, then the planes intersect in a line.

if there is no solution then the planes do not intersect. (Planes are skew; is that the right terminology here?)

Is what I said so far correct, and are there any other scenarios? Also, say you have a set of 3 equations and you're told there is no solution at all; is there any propery of the matriz that corresponds to the set of equations that tells you that there is no solution? i.e.) Something similar to Det. = 0 to imply no unique solution?

Cheers.

Reply 1

Expression

14

Nima

I don't understand what 3 planes defined by 3 equations are supposed to represent geometrically.

Is this right - If there is a unique solution to the 3 equations then this means that the 3 planes intersect a single point in common.

If there is no unique solution i.e.) The determinant of the matrix that corresponds to the set of 3 equations = 0, then the planes intersect in a line.

if there is no solution then the planes do not intersect. (Planes are skew; is that the right terminology here?)

Is what I said so far correct, and are there any other scenarios? Also, say you have a set of 3 equations and you're told there is no solution at all; is there any propery of the matriz that corresponds to the set of equations that tells you that there is no solution? i.e.) Something similar to Det. = 0 to imply no unique solution?

Cheers.

The only other one is where there are infinately many solutions (x*0 = 0), in which case we form a SHEAF.

Example
For the case of a=5, b=-16 solve the following Simultaneous Equations:

2x - y - z = 16 (*)
3x + y - 4z = -1 (**)
4x + ay - 9z = b (***)

(*) + (**) => 5x - 5z = 5 ==> x - z = 1, x = z+1.

3(*) - 2(**) ==> -5y + 5z = 20 ===> z - y = 4, y = z+4.

Sub x = z+1, and y= z-4 into (***),

4(z+1) + a(z+4) - 9z = b
=> (a-5)z = 4a + b - 4

Now,

Sub. a=5, b=-16.

(5-5)z = 4(5) + (-16) -4
0z = 20-16-4
0z = 0.

No restriction on z, equations will have infinately many solutions.
Hence the 3 planes form a SHEAF.

Reply 2

username9816

OP

14

Expression

The only other one is where there are infinately many solutions (x*0 = 0), in which case we form a SHEAF.

What's a sheaf?

And are you sure about that? Coz the mark scheme says that "there are infinitely many solutions so the planes intersect in a line."

Reply 3

Expression

14

Sheaf of Planes - good diagram too.

I should add to the notes above that to geometrically represent a point, a sheaf or a prism, that no two of the planes should be parallel to eachother.

----

3.5 Planes

The solution sets for intersection of three, non-parallel planes can be categorized as one of the following cases:

A. Unique Solution
Three planes intersect at one point

B. Infinately many solutions
The planes form a sheaf

C. No solutions
Planes form a hollow prism.

Reply 4

Spenceman_

Might not be included in every syllabus but it's also worth mentioning the case where they give you equations which aren't independent and you get two or even all three of the planes being parallel. :wink:

Reply 5

Expression

14

Nima

And it is coz the math link u posted says so. :biggrin:

Woohooo, glad that's cleared up.

The diagram does indeed show it with the planes through a line, so with enough planes you would get a nice axis (the "line") of the sheaf.

:biggrin:

You weren't wrong, I'd just never heard it described as that.

Reply 6

posh_git

1

i don't get the whole numerical methods thing lol as one of my friends said "thats stupid....why can't they do it the normal simultaneos equations way instead of keeping the top one the same everytime?" lol!

Three Planes Defined By 3 Equations (P6 Stuff)

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