Exercises: Perpendicularity and parallelism

Imagine you have a shoebox (shaped like a rectangular prism) on your desk. Looking at the edges of the lid:

1. a) State two edges that are parallel to each other.
2. b) State two edges that are perpendicular and intersect at a vertex.

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.

1. a) What is the relative position of the column and the floor? Justify your answer.

1. 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?

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:

1. a) What is the relative position of these two paths in space called?
2. b) Are these lines coplanar? Explain why.

Look at the walls of your room.

1. a) What is the relative position of two opposite walls?
2. 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.

1. a) If we open the book completely until it lies flat on the table, what is the size of that dihedral angle?
2. 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?

Look at the following cube and identify:

1. a) An edge that is parallel to the plane of the bottom base.
2. b) An edge that is perpendicular to the plane of the bottom base.

Imagine a soccer goal. Let us analyze its posts as lines:

1. a) What is the relative position of the two vertical posts?
2. b) What is the relative position of the crossbar (horizontal post) with respect to each of the vertical posts?

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 $r$):

1. a) What is the relation between line $r$ and the floor plane $\pi$?
2. b) What is the relative position of line $r$ with respect to the side wall plane $\beta$?
3. 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.

1. a) What is the name of each of the three regions of space bounded by two consecutive panels?
2. 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?
3. c) If a fly lands exactly on the central axis, how many faces of the dihedral angles does it lie on?

Consider a right prism with a regular hexagonal base. Analyse the relationship between its elements:

1. a) How many lateral faces are perpendicular to the bottom base?
2. b) How many pairs of lateral faces are parallel to each other?
3. 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?

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.

1. a) Is it possible for this ray to pass through the curved surface of the cone?
2. 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 $\pi_2$, perpendicular to the first, also intersects the sphere, passing through its center.

1. a) Describe the relative position of the line of intersection of both planes with respect to the sphere.
2. 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.