# 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

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