What Is the Necessary Condition for Three Planes to Intersect Along a Line?

[canadian_beef]

Homework Statement

Find a necessary condition for the three planes given below to have a line of intersection.

-x +ay+bz=0
ax-y+cz=0
bx+cy-z=0

Homework Equations

in order to get a line of intersection between the planes..i know i need one line of the matrix to be [0 0 0|0]

The Attempt at a Solution

well here's the attempt..and its wrong

[ -1 a b | 0
a -1 c | 0
b c -1| 0 ]

=>

[-1 a b | 0
0 (a^2-1) ba+c | 0 (aRow1 + Row2)
0 (ab+c) b^2+1 | 0 ] (brow1 + Row 2)


=>

[ -1 a b | 0
0 a^2 -1 ba+c |0
0 0 2abc +c^2 - a^2 + b^2 +1) |0 ] (ab+c row2- a^2-1 Row1)


then what i did ..by inspection i made 2abc+c^2 -a^2 +b^2 +1 = 0 by letting a=b=1, and c=-1...

but that doesn't work becasue that owuld make plane 1 and 2 the same plane.

i need help

thanks

 

[AKG]

The third row in your original matrix should be "b -c -1 | 0" not "b c -1 | 0".

 

[canadian_beef]

my bad..edited...i mistyped the question

but still need help

 

[AKG]

In your last matrix, the 3rd element of the third row is "2abc +c² - a² + b² +1" but then you start looking at the equation "2ab + c² - a² + b² +1 = 0".

 

[canadian_beef]

In your last matrix, the 3rd element of the third row is "2abc +c² - a² + b² +1" but then you start looking at the equation "2ab + c² - a² + b² +1 = 0".

another typo on my part i have that c there

 

[Hurkyl]

then what i did ..by inspection i made 2abc+c^2 -a^2 +b^2 +1 = 0 by letting a=b=1, and c=-1...

but that doesn't work becasue that owuld make plane 1 and 2 the same plane.

So? You weren't asked to find a sufficient condition, you were asked to find a necessary condition.

Incidentally, you have either the polynomial wrong, or the matrix wrong: I think determinants are a simpler approach to the problem than Gaussian elimination.

 

[canadian_beef]

Im not sure how to do it the dertiminant way. I do not think my math is wrong so far.

Help

 

[AKG]

"a=b=1, c=-1" is a sufficient condition, not a necessary condition. In fact, "2abc + c² - a² + b² +1 = 0" is also just a sufficient condition, not a necessary condition, since it isn't necessary for the third line to be all zeroes (the second line could be all zeroes).

 

[canadian_beef]

"a=b=1, c=-1" is a sufficient condition, not a necessary condition. In fact, "2abc + c² - a² + b² +1 = 0" is also just a sufficient condition, not a necessary condition, since it isn't necessary for the third line to be all zeroes (the second line could be all zeroes).

ok thanks

what would be an example as a necessary conditon and how would i go about finding it

 

[HallsofIvy]

If the matrix of coefficients were invertible then the only simultaneous solution to the three equations would be (0, 0, 0), the POINT of intersection of the three planes. In order that the three planes intersect in a line it is necessary that the matrix not be invertible: in other words that the determinant be 0. Find the determinant and set it equal to 0.