Relative Position of Lines and Planes in Space

Submitted by vladimir.arzt on Sat, 03/16/2024 - 20:52
Project ID: 
7300000024
Accepted: 
SubArea: 
Space geometry
Type: 
Scrolling
Layout: 
8x8
Question: 
Match each pair of equations of lines/planes with information about their relative position.
Questions Title: 
Equations of lines/planes
Answers Title: 
The relative position of lines/planes
Question 1: 
x=3−ty=−2−2tz=1+t } t∈Rx=−4+sy=−7+3sz=−1−2s } s∈R
Answer 1: 
The lines intersect.
Question 2: 
x=3−2ty=−2−3tz=2+t } t∈R3x−4y+2z+3=0
Answer 2: 
The line and the plane intersect in a point.
Question 3: 
x−2y+z−2=03x−4y+2z+3=0
Answer 3: 
The planes intersect in a line.
Question 4: 
x=−4+4ty=1+3tz=−1−2t } t∈R2x−4y−2z+3=0
Answer 4: 
The line and the plane are parallel, the line does not lie in the plane.
Question 5: 
x=3−2ty=−2−4tz=2+t } t∈Rx=−4+sy=−7+2sz=1−2s } s∈R
Answer 5: 

The lines are skew,

i.e., they do not intersect,

and there is no plane that contains them.

Question 6: 
x=3+2ty=2+3tz=2+2t } t∈R3x−4y+3z−7=0
Answer 6: 
The line lies in the plane.
Question 7: 
x=3−2ty=−2−4tz=2+6t } t∈Rx=−4+sy=−7+2sz=1−3s } s∈R
Answer 7: 
The lines are parallel, they have no point of intersection.
Question 8: 
2x−6y+4z−3=03x−9y+6z−5=0
Answer 8: 
The planes are parallel, they do not intersect.