# Relative positions

A. TWO PLANES  - If rk(A)=rk(A*)=1 → coincident planes - If rk(A)=1≠2=rk(A*) → parallel planes - If rk(A)=rk(A*)=2 → intersecting planes Example 1:  rk(A)=1≠2=rk(A*) → parallel planes

Example 2:  rk(A)=rk(A*) =2 → intersecting planes

B. A PLANE AND A STRAIGHT LINE  - If rk(A)=rk(A*)=2 →  straight line into the plane - If rk(A)=2≠3=rk(A*) → parallel ones - If rk(A)=rk(A*)=3 → intersecting ones Example 3:  rk(A)=rk(A*)=2 → line into the plane

Example 4:  rk(A)=rk(A*) =3 → intersecting ones

C. TWO STRAIGHT LINES  - If rk(A)=rk(A*)=2 →  coincident lines - If rk(A)=2≠3=rk(A*) →  parallel lines - If rk(A)=rk(A*)=3 → intersecting lines - If rk(A)=3≠4=rk(A*) → skew lines Example 5:  rk(A)=3≠4=rk(A*) → skew lines

Example 6:  rk(A)=rk(A*) =3 → intersecting ones

D. THREE PLANES  - If rk(A)=rk(A*)=1 → coincident planes - If rk(A)=1≠2=rk(A*) → Two coincident and one parallel or three parallel planes  - If rk(A)=rk(A*)=2 → Either they intersect at a line or there are two coincident and the other intersects at a line  - If rk(A)=2≠3=rk(A*) → Either there are two parallel ones and another one intersecting or they form a triangular prism  - If rk(A)=rk(A*)=3 → They intersect at a point Example 7:  rk(A)=2≠3rk(A*)→ they from a prism

Example 8:  rk(A)=rk(A*) =3 → intersecting ones

Exercises:

1.- Study the relative position of the plane π: x + 2y - 1 = 0 and the straight line r: 2.- Study the relative position of the lines r and s, where: Solutions: 1) Parallel ones; 2) Intersecting lines