How many planes pass through #3# distinct points in #3# dimensional space?
1 Answer
If the points are colinear then an infinite number of planes can be made to pass through them. If three distinct points are non-colinear then exactly one plane passes through them.
Explanation:
Let us look at the general case.
The general equation of a plane is
For a given plane, this equation is unique only up to a constant factor. That is, the same plane is also described by the equation:
Let the three points be
Then we have a system of three linear equations:
In the general case, this is a little painful to deal with using substitution and/or elimination, so permit me to show how to do it using matrix arithmetic.
Let
Then our three equations become:
If we can find
Let:
The determinant of
If
How can this go wrong?
If the points are not distinct, then two of the rows of
If the points are colinear, then
If the plane also passes through
This last case is the one where