State and Derive Theorem for Parallel and Perpendicular Axes

Answer:

• A line is perpendicular to a plane if and only if the line is perpendicular to every vector in the plane
• A plane is perpendicular to another plane if it is perpendicular to every line in the other plane
• Parallelism and perpendicularity are transitive relationships

Theorem for parallel and perpendicular axes.

Similarly, a line is parallel to a plane if and only if the line is parallel to every vector in the plane. These relationships also hold for planes: a plane is parallel to another plane if and only if it is parallel to every vector in the other plane, and a plane is perpendicular to another plane if and only if it is perpendicular to every line in the other plane. In other words, parallelism and perpendicularity are transitive relationships.

For example, consider the line l and plane P:

Since l is perpendicular to every vector in P, we can say that l is perpendicular to P. Similarly since l is parallel to every vector in P, we can say that l is parallel to P.

Now consider the plane P and plane Q

Since P is perpendicular to every line in Q, we can say that P is perpendicular to Q. Similarly since P is parallel to every vector in Q, we can say that P is parallel to Q.

Thus, we can see that the relationships of parallelism and perpendicularity are indeed transitive.