Perpendicularity and parallelism
Look around your bedroom: the floor and the ceiling never meet, two walls join at a corner, and the edge of the table and a vertical corner of the wall neither cross nor are parallel. In space there are more possibilities than in the plane, and it helps to have a name for each one.
Positions of two lines
In the plane, two lines either intersect or are parallel. In space a third possibility appears: they can be skew (they neither intersect nor are parallel, because they do not lie in the same plane).
| Position | Do they intersect? | Do they lie in the same plane? |
|---|---|---|
| Parallel | No | Yes |
| Intersecting | Yes, at a point | Yes |
| Skew | No | No |
If two intersecting lines form 90°, they are perpendicular.
Two aeroplanes are flying in the sky. Aeroplane A is flying North at an altitude of $10.000$ metres. Aeroplane B is flying East at an altitude of $8.000$ metres. Given…
Two aeroplanes are flying in the sky. Aeroplane A is flying North at an altitude of $10.000$ metres. Aeroplane B is flying East at an altitude of $8.000$ metres. Given that their flight paths do not intersect and are not parallel:
- a) What is the relative position of these two paths in space called?
- b) Are these lines coplanar? Explain why.
Imagine a football goal. Let us analyse its posts as lines: a) What is the relative position of the two vertical posts? b) What is the relative position of the crossbar…
Imagine a football goal. Let us analyse its posts as lines:
- a) What is the relative position of the two vertical posts?
- b) What is the relative position of the crossbar (horizontal post) with respect to each of the vertical posts?
Lines and planes
A line relative to a plane can be contained in it, be parallel (it never touches it) or be intersecting (it meets the plane at one point). If it intersects and forms 90° with every line of the plane that passes through that point, it is perpendicular to the plane: like a column standing on the floor.
In the school gym, a straight column (line $r$) meets the floor (plane $\pi$) at a point, which is its base. On the floor there are two distinct lines, $s$ and $t$, that…
In the school gym, a straight column (line $r$) meets the floor (plane $\pi$) at a point, which is its base. On the floor there are two distinct lines, $s$ and $t$, that pass through the base of the column, and it is known that the column forms a right angle with each of them.
- a) What is the relative position of the column and the floor? Justify your answer.
- b) If we draw any other line on the floor that passes through the base of the column, what angle will it form with the column?
Look at the following cube and identify: a) An edge that is parallel to the plane of the bottom base. b) An edge that is perpendicular to the plane of the bottom base.
Look at the following cube and identify:
- a) An edge that is parallel to the plane of the bottom base.
- b) An edge that is perpendicular to the plane of the bottom base.
Two planes and dihedral angles
Two planes can be parallel (floor and ceiling) or intersecting (two walls): in that case they meet along a line. The region between two half-planes that share that line is called a dihedral angle, like an open book.
Look at the walls of your room. a) What is the relative position of two opposite walls? b) What is the line called where two adjacent walls intersect?
Look at the walls of your room.
- a) What is the relative position of two opposite walls?
- b) What is the line called where two adjacent walls intersect?
When you open a maths textbook and leave it resting on the table, the two pages form a dihedral angle. a) If we open the book completely until it lies flat on the table,…
When you open a maths textbook and leave it resting on the table, the two pages form a dihedral angle.
- a) If we open the book completely until it lies flat on the table, what is the size of that dihedral angle?
- b) If the book is half-open and the pages form an angle of $90^\circ$, how are the planes of the two pages related to each other?
Imagine that the floor of your room is plane $\pi$ and one of the side walls is plane $\beta$. If you install a fibre optic cable along the corner where the ceiling…
Imagine that the floor of your room is plane $\pi$ and one of the side walls is plane $\beta$. If you install a fibre optic cable along the corner where the ceiling meets the front wall (line $r$):
- a) What is the relation between line $r$ and the floor plane $\pi$?
- b) What is the relative position of line $r$ with respect to the side wall plane $\beta$?
- c) If we draw a vertical line on the side wall (line $s$), will lines $r$ and $s$ intersect? Justify whether they are perpendicular.
Back to the beginning: In your bedroom, the floor and the ceiling are parallel planes, two walls are intersecting planes that meet along an edge, and the edge of the table and the vertical corner are two skew lines.