Space geometry

1103212902

Level: 
C
A cube ABCDEFGH with an edge length of 2 units is placed in a coordinate system (see the picture). Let S be the midpoint of the face ABFE, and let K and L be the midpoints of edges DH and CG consecutively. Find the standard equation of a plane α passing through the points A, B and L, and calculate the distance of the point S from α.
α:x+2z−2=0; |Sα|=255
α:x+2z−2=0; |Sα|=233
α:x+2y−2=0; |Sα|=255
α:x+2y−2=0; |Sα|=233

1003188803

Level: 
A
A plane ρ is defined by the point A=[3;1;1] and a straight line p defined by the following parametric equations: p:x=4+4t,y=−1−2t,z=1+t; t∈R Find the parametric equations of the plane ρ.
ρ:x=4+4t+s,y=−1−2t−2s,z=1+t; t,s∈R
ρ:x=4+4t+3s,y=−1−2t+s,z=1+t+s; t,s∈R
ρ:x=3+4t+4s,y=1−2t−s,z=1+t+s; t,s∈R
ρ:x=3+4t−4s,y=1−2t+2s,z=1+t−s; t,s∈R

1003188802

Level: 
A
Find the missing coordinates of the pointsM=[2;m;0] and N=[0;3;n] so that they lie on a plane ρ defined by the following parametric equations: ρ:x=4+2s,y=−1−2t,z=1+t+s; t,s∈R Choose the option in which values of both m and n are correct.
m=−1, n=−3
m=−1, n=3
m=1, n=−3
m=1, n=3

1003188801

Level: 
A
We are given points A=[2;4;0], B=[4;−1;1] and C=[0;1;1]. From the following list, choose the parametric equations which represent a plane ρ defined by the points A, B, and C.
ρ:x=4+2t+2s,y=−1−t−5s,z=1+s; t,s∈R
ρ:x=4+4t+2s,y=−1−2t−5s,z=1+t+s; t,s∈R
ρ:x=2t+4s,y=1−t−2s,z=1; t,s∈R
ρ:x=2t−2s,y=1−5t+5s,z=1+t−s; t,s∈R

1103188706

Level: 
A
We are given points A=[2;4;0] and B=[4;7;6]. Find parametric equations of a line q, which is the orthogonal projection of the line AB into the coordinate plane xy.
p:x=4+2t,y=7+3t,z=0; t∈R
p:x=2+4t,y=4+7t,z=6t; t∈R
p:x=4+2t,y=7+3t,z=6; t∈R
p:x=2−2t,y=4−3t,z=−6t; t∈R