Space geometry

1003233607

Level: 
C
Determine the relative position of three planes: α: 2x+y+9z−18=0,β: x+3y+2z+16=0,γ: x+2y+3z+6=0.
Planes α, β and γ intersect in a straight line.
Each of the two planes are intersecting and the lines of intersection are three different lines parallel to each other.
All three planes intersect at just one point.

1003233605

Level: 
C
We are given skew lines p and q. p:x=1−t,q:x=1−2s,y=1+t,y=s,z=3+2t; t∈R,z=3+3s; s∈R. Find parametric equations of a straight line r, that is intersecting both lines p and q and lying in the plane x+2y−z+2=0.
r:x=−1+2m,y=3−3m,z=7−4m; m∈R
r:x=−1+m,y=3+3m,z=7−m; m∈R
r:x=−1+3m,y=3+2m,z=7+5m; m∈R
r:x=−1+m,y=3−m,z=7+m; m∈R

1103233603

Level: 
C
Let ABCDEFGH be a cube with an edge length of 1 placed in the rectangular coordinate system. In the cube a regular tetrahedron ACHF is highlighted (see the picture). Find the angle between its faces and round the number to the nearest minute.
70∘32′
54∘44′
45∘
51∘4′

1103233602

Level: 
C
Let ABCDEFGH be a cube with an edge length of 1 placed in the rectangular coordinate system. In the cube a regular tetrahedron ACHF is highlighted (see the picture). Find the distance between the opposite edges of this tetrahedron. Hint: A tetrahedron’s opposite edges lie on skew lines. Their distance is the same as the distance of the midpoint of one edge from the opposite edge.
1
3
32
52