Matrix Intersection of Three planes?

Question

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




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]




well heres 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 doesnt work becasue that owuld make plane 1 and 2 the same plane.

i need help

thanks

what i have done is found how to do a sufficient condition...how do just find a general necessary condition

Answer 1

The intersection of two planes is a line. For three planes to intersect at a line. the linemust, of course, be the same one that the two intesect at. In short, the three planes cannot be independent because the constraint forces the intersection. Since they are not independent, the determineant of the coefficient matrix must be zero so:

| -1 a b |
| a -1 c | = 0
| b c -1 |

-1 + abc + abc - (-b^2 -c^2 - a^2) = 0

a^2 + b^2 + c^2 + 2abc = 1 is your necessary condition.

You could pick values for a, b, and c and those values might make 2 or three of the planes the same. That doesn't mean your solution is wrong, just that you picked an unfortunate set of numbers.

For example, pick a = 1.7, b = -2.5, then c could be either 7.4 or 1.1 giving two different sets of three diferent planes.

Answer 2

ok that first matrix that you have.... what you do is first of all remove those zeros.... then you have a coefficient matrix that is 3x3. Now that condition that you are looking for is one that will make those 3 planes intersect to form a line. Well as it turns out those 3 planes intersecting will also intersect at where 2 of those planes intersect.
Ok what you want to do is to take that coefficient matrix and find the values of a, b, and c where the determinite of that matrix will = 0.
Find the formula of finding a matrix of a 3x3 matrix (it is kind of long) and set that equal to zero and find the possible values of a, b and c. That will give you your criteria for those 3 planes to intersect at a line.