K. Nicole Clark Math 311-200

URL: people.tamu.edu/~knc8928/5.1.18.pdf

ญญ**5.1.18**

Consider affine subspaces in *R*^{3} (lines and
planes that do not necessarily pass through the origin). Use Theorem 6 to classify the possible
intersections and unions of

(a) a plane and a line;

(b) two planes.

HINT: If the sets intersect at all, you can choose the origin of coordinates to be a point in the intersection.

**Solution**

(a) A plane and a line can intersect at a point, the line can lie in the plane, or they can be parallel and not intersect at all. As is obvious, a line is one-dimensional and a plane is two-dimensional. We are going to assume the origin falls anywhere along the intersections.

** Case I**: A
plane and a line intersect at a point.

__Intersection__: dim(L_{}P) = 0 this is because the intersection is at a point,
which has a dimension of zero. (Having a
point as an intersection is also called a *direct
sum*).

__Union__*Linearity*,
we know that

_{}

The second equation is used when the intersection is a direct sum, as defined above.

Thus, in this case,

_{}

** Case II**: A
line lies in a plane.

__Intersection__: dim(L_{}P) =1 - In this case, the intersection of the two will be simply
the line itself, which has by definition a dimension of one.

__Union__

_{} = 1 + 2 -1 = 2

This is obvious from intuition as well, as the union of a plane and a line that lies in that plane will be the plane itself, which by definition has a dimension of two.

** Case III**:
The line and the plane never intersect.

In this case, it is impossible for both the line and the plane to pass through the origin, thus Theorem 6 does not apply.

(b) Two planes can intersect at a line, lie in the same plane, or be parallel and not intersect at all. We are again assuming that the origin falls somewhere along the intersection

** Case I**:
Two planes intersect along a line.

__Intersection__: _{}- This makes sense, as the dimension of a line is one.

__Union__

_{} = 2 +2 1 = 3

** Case II: **Two
planes lie in the same plane.

__Intersection: __This intersection is just a plane,
having a dimension of two by definition.

_{}2

__Union__

_{} = 2 +2 2 = 2

This makes sense, as the union of two planes lying in the same plane is just a plane.

** Case III: **The
two planes never intersect.

In this case, it is impossible for both planes to pass through the origin, thus Theorem 6 does not apply.