Intersection of 2 planes (HL)
See also intersection of 3 planes (HL)
Two planes can be identical (the same), parallel, or intersecting.
Example: Let’s find the intersection between
Contents
Method using cross product
The line of intersection is perpendicular to both normals and .
step 1: direction vector
The direction vector of the line is .
step 2: a point
For the component of that is closest to , but is not , guess the corresponding component of a point.
In our example, since is the closest component to , let’s guess . (You can also guess , for instance, but is often easier)
This means
which has solution ,
The final answer is
Method using row reduction
In this context, scalar multiplication means multiplying the row by a non-zero constant.
The rules are
- Each elementary row operation uses some or all rows from the previous step.
- Two rows may be swapped.
- Each changed row must involve the corresponding row from previous step, unless two rows are swapped. Eg the new row 2 must involve a non-zero multiple of the old row 2, unless swapping row 2 with another.
- Each row can become a scalar multiple of itself, and/or added a scalar multiple of any other row(s). Eg new row 1 can be times old row 1 plus half times old row 3.
The goal is to get the it to look like
where at most a single column on the left side does not contain a one and a zero.
Our starting point is
Generally speaking we focus on getting one column into the desired state, in a left-to-right order.
We can first subtract row 2 from row 1
Subtract twice of row 1 from row 2
Subtract row 2 from row 1
This corresponds to the system
can be used to parametrize both and , as it appears in both equations.
Let
This means
which are the same equations as before, but in a different form. In both cases we assumed for the point. Had we used different assumptions, we would end up with different points.