- #1
nobahar
- 497
- 2
Hello!
I have a quick question regarding the intersection of three planes if the determinant is 0.
If there are solutions, there will be an infinite number of solutions. One of the equations for the plane can be ignored as it is a linear combination of the other two, and can be ignored for the purposes of finding a solution (as far as I am aware).
Okay, this is where I think I may be wrong. The components of a normal vector to the plane are not like a position vector, specifying some point a certain distance from the origin. Because a normal vector can be "moved around": it's dot product with any vector in the plane yields 0. If this is the case, the cross product of two normal vectors (one for each of the two planes), yields some vector with specific values. Is it true that this vector, V, and it's scalar multiples are not necessarily the solutions? Because there is nothing in the normal vectors specifying a location; and the vector that is a product of the cross product of the two normal vectors also does not have any 'information' specifying its location. It will be parallel to the line, but not necessarily on the line. I have seen an example of finding the solutions to the intersection of three planes but no more information was needed (i.e. scalar multiples of the cross product of two normal vectors were the solutions), but the trivial solution (x=y=z=0) was a, erm, solution! Therefore it passes through the origin, and I think that the origin can be used to specify the position of the vector V.
Any help appreciated, sorry if it's not too clear...
I have a quick question regarding the intersection of three planes if the determinant is 0.
If there are solutions, there will be an infinite number of solutions. One of the equations for the plane can be ignored as it is a linear combination of the other two, and can be ignored for the purposes of finding a solution (as far as I am aware).
Okay, this is where I think I may be wrong. The components of a normal vector to the plane are not like a position vector, specifying some point a certain distance from the origin. Because a normal vector can be "moved around": it's dot product with any vector in the plane yields 0. If this is the case, the cross product of two normal vectors (one for each of the two planes), yields some vector with specific values. Is it true that this vector, V, and it's scalar multiples are not necessarily the solutions? Because there is nothing in the normal vectors specifying a location; and the vector that is a product of the cross product of the two normal vectors also does not have any 'information' specifying its location. It will be parallel to the line, but not necessarily on the line. I have seen an example of finding the solutions to the intersection of three planes but no more information was needed (i.e. scalar multiples of the cross product of two normal vectors were the solutions), but the trivial solution (x=y=z=0) was a, erm, solution! Therefore it passes through the origin, and I think that the origin can be used to specify the position of the vector V.
Any help appreciated, sorry if it's not too clear...
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