Perpendicularity and Parallelism
Look at the ceiling of your classroom at school: the floor and the ceiling never touch, two walls meet at a corner, and the edge of the board and the vertical corner of the wall neither intersect 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 are not in the same plane).
| Position | Do they intersect? | Are they in the same plane? |
|---|---|---|
| Parallel | No | Yes |
| Intersecting | Yes, at one point | Yes |
| Skew | No | No |
Two lines that lie in the same plane are called coplanar. Skew lines are not coplanar. If two intersecting lines form $90^\circ$, they are perpendicular ($r \perp s$); if they are parallel, we write $r \parallel s$.
Imagine you have a shoebox (shaped like a rectangular prism) on your desk. Looking at the edges of the lid: a) State two edges that are parallel to each other. b) State…
Imagine you have a shoebox (shaped like a rectangular prism) on your desk. Looking at the edges of the lid:
- a) State two edges that are parallel to each other.
- b) State two edges that are perpendicular and intersect at a vertex.
Two airplanes are flying in the sky. Airplane A is flying North at an altitude of $10,000$ meters. Airplane B is flying East at an altitude of $8,000$ meters. Given that…
Two airplanes are flying in the sky. Airplane A is flying North at an altitude of $10,000$ meters. Airplane B is flying East at an altitude of $8,000$ meters. 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 soccer goal. Let us analyze 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 soccer goal. Let us analyze 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 to it (it never touches it), or be intersecting (it crosses it at one point). If it intersects the plane and forms $90^\circ$ 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.
Consider a right prism with a regular hexagonal base. Analyse the relationship between its elements: a) How many lateral faces are perpendicular to the bottom base? b)…
Consider a right prism with a regular hexagonal base. Analyse the relationship between its elements:
- a) How many lateral faces are perpendicular to the bottom base?
- b) How many pairs of lateral faces are parallel to each other?
- c) If we draw the diagonal of a lateral face and a base edge that shares no vertex with that diagonal, what is the relative position of the lines containing them?
Two Planes and Dihedral Angles
Two planes can be parallel (floor and ceiling) or intersecting (two walls): then they meet along a line. The region between two half-planes that share that line is called a dihedral angle, like an open book. The common line is the edge of the dihedral angle and each half-plane is a face.
A dihedral angle is measured with a plane angle: the one formed by two lines, one in each face, perpendicular to the edge at the same point. If it measures $90^\circ$, the planes are perpendicular (two walls forming a corner). If the book is laid completely open on the table, the dihedral angle measures $180^\circ$.
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 math 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 math 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 fiber 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 fiber 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.
A revolving door in a shopping center has three identical glass panels that intersect along a central vertical axis. a) What is the name of each of the three regions of…
A revolving door in a shopping center has three identical glass panels that intersect along a central vertical axis.
- a) What is the name of each of the three regions of space bounded by two consecutive panels?
- b) If the dihedral angle between the first and second panels is $120^\circ$, and the three panels are symmetrically distributed, what are the sizes of the other two dihedral angles?
- c) If a fly lands exactly on the central axis, how many faces of the dihedral angles does it lie on?
Solids and Cross Sections
These ideas help you reason about solids. The height of a prism or a pyramid is the segment perpendicular to the plane of the base. In a sphere, the radius that reaches a point of tangency is perpendicular to the tangent plane, and the line where two planes through it meet is studied in a drawing of the cross section.
A point light source is located at the center of the base of a right cone of radius $R = 5$ cm and height $H = 12$ cm. The light source emits a light ray that is…
A point light source is located at the center of the base of a right cone of radius $R = 5$ cm and height $H = 12$ cm. The light source emits a light ray that is perpendicular to one of the slant heights (generators) of the cone.
- a) Is it possible for this ray to pass through the curved surface of the cone?
- b) Calculate the distance from the light source to the point on the slant height where the ray is perpendicular. To do this, picture the right triangle formed by the axial cross-section of the cone.
A sphere has a radius of $13$ cm. A plane $\pi_1$ intersects the sphere at a distance of $5$ cm from its center, forming a circle $C_1$. A second plane $\pi_2$,…
A sphere has a radius of $13$ cm. A plane $\pi_1$ intersects the sphere at a distance of $5$ cm from its center, forming a circle $C_1$. A second plane $\pi_2$, perpendicular to the first, also intersects the sphere, passing through its center.
- a) Describe the relative position of the line of intersection of both planes with respect to the sphere.
- b) What is the minimum distance between the boundary of circle $C_1$ and plane $\pi_2$? Use a diagram of the circular cross-section to justify your answer.
Back to the beginning: in your classroom, the floor and the ceiling are parallel planes, two walls are intersecting planes that meet along an edge, and the edge of the board and the vertical corner are two skew lines.