Parallel and Perpendicular Lines and Planes in Space
Look around your bedroom. The floor and the ceiling never touch. Two walls meet along a corner. And the front edge of your desk (if it's pulled away from the wall) and the vertical corner of the room neither cross nor run side by side. In a flat drawing, two lines either cross or are parallel. In space there are more possibilities, and each one needs a name and a test.
Two lines in space
In a plane, two different lines either intersect or are parallel. In space there is a third option: the lines can be skew. They never meet, but they are not parallel either, because they do not lie in the same plane.
In the cube, $A, B, C, D$ are the bottom vertices and $E, F, G, H$ are directly above them. Edge $AB$ is blue. Each other colored edge shows one relationship with $AB$.
| Position | Do they meet? | In the same plane (coplanar)? |
|---|---|---|
| Parallel | No | Yes |
| Intersecting | Yes, at one point | Yes |
| Skew | No | No |
Two lines are coplanar exactly when they are parallel or intersecting. Skew lines are the only pair that is not coplanar.
Two intersecting lines that form a $90^\circ$ angle are perpendicular. Skew lines can also be perpendicular. To measure the angle between skew lines, slide one of them (keeping its direction) until it hits the other, then measure the angle. If that angle is $90^\circ$, the lines are perpendicular skew lines.
Imagine you have a shoebox (rectangular prism shape) on your table. If we look at the edges of the lid: a) State two edges that are parallel to each other. b) State two…
Imagine you have a shoebox (rectangular prism shape) on your table. If we look 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 flies North at a height of $10.000$ feet. Airplane B flies East at a height of $8.000$ feet. Knowing that their paths do…
Two airplanes are flying in the sky. Airplane A flies North at a height of $10.000$ feet. Airplane B flies East at a height of $8.000$ feet. Knowing that their 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.
In the design of a modern cube-shaped shelving unit (as shown in the figure), some of its edges have been numbered. Identify the relative position in space of the…
In the design of a modern cube-shaped shelving unit (as shown in the figure), some of its edges have been numbered. Identify the relative position in space of the following pairs of lines containing these edges:
- a) The lines passing through edges $r$ and $s$.
- b) The lines passing through edges $s$ and $t$.
- c) The lines passing through edges $r$ and $p$.
- d) State an edge that is perpendicular to the top face and does not intersect edge $s$.
Lines and planes
A line and a plane can be in one of these positions:
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Contained: every point of the line is in the plane. (If a line touches a plane in two points, it is contained.)
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Parallel: they never meet.
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Intersecting: they meet in exactly one point.
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Perpendicular: intersecting, and the line forms $90^\circ$ with every line of the plane that passes through the intersection point. Think of a column standing on the gym floor.
To decide if a line is parallel to a plane, use this test.
In the school gym, one of the vertical columns (line $r$) meets the floor (plane $\pi$). a) What is the relative position of the column and the floor? b) If we draw any…
In the school gym, one of the vertical columns (line $r$) meets the floor (plane $\pi$).
- a) What is the relative position of the column and the floor?
- b) If we draw any line on the floor passing through the base of the column, what angle will it form with it?
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.
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 that runs along the corner where the…
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 that runs along the corner where the ceiling and the front wall meet (line $r$):
- a) What is line $r$ like with respect to 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.
Tests for perpendicularity
Checking all the lines of a plane would take forever. These rules save the work.
The test works for the same reason a flagpole stays straight: two guy wires on different sides hold it upright. A single wire would let it lean sideways.
Imagine a soccer goal. Let's 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's 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?
Two lines in space, $a$ and $b$, cross without intersecting. If line $a$ is parallel to a horizontal plane $\pi$ and line $b$ is perpendicular to that same plane $\pi$:…
Two lines in space, $a$ and $b$, cross without intersecting. If line $a$ is parallel to a horizontal plane $\pi$ and line $b$ is perpendicular to that same plane $\pi$:
- a) What angle do the directions of lines $a$ and $b$ form?
- b) Is it possible to find a plane that contains both lines at the same time? Why?
- c) Can we state that the two lines are perpendicular even though they have no points in common?
In a regular quadrangular pyramid (square base), the top vertex is called the apex. a) Justify why the axis of the pyramid (the line joining the apex to the center of…
In a regular quadrangular pyramid (square base), the top vertex is called the apex.
- a) Justify why the axis of the pyramid (the line joining the apex to the center of the base) is perpendicular to any diagonal of the square base.
- b) Is there any lateral edge that is parallel to a base edge? Reason your answer.
- c) What are the planes containing two opposite lateral faces like? Are they intersecting or parallel?
Two planes
Two different planes are either parallel (no common point, like floor and ceiling) or intersecting. If they intersect, they meet in a whole line, never in a single point (like two adjacent walls).
Why does fact 1 work? Both lines lie in the cutting plane $\gamma$, so they are coplanar. One lies in $\alpha$ and the other in $\beta$, which never meet, so the lines never meet. Coplanar lines that never meet are parallel.
Look at the walls of your room. a) What is the relative position of two opposite walls? b) What is the line formed when two adjacent walls intersect called?
Look at the walls of your room.
- a) What is the relative position of two opposite walls?
- b) What is the line formed when two adjacent walls intersect called?
Draw three planes ($\pi_1, \pi_2, \pi_3$) in your notebook that meet the following conditions: 1. $\pi_1$ and $\pi_2$ are parallel to each other. 2. $\pi_3$ is…
Draw three planes ($\pi_1, \pi_2, \pi_3$) in your notebook that meet the following conditions:
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$\pi_1$ and $\pi_2$ are parallel to each other.
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$\pi_3$ is perpendicular to $\pi_1$.
What is the relative position between $\pi_3$ and $\pi_2$?
Given three planes in space $\pi_1, \pi_2$ and $\pi_3$, we know that $\pi_1$ is parallel to $\pi_2$ and that $\pi_3$ intersects $\pi_1$ forming a line $r$. a) What will…
Given three planes in space $\pi_1, \pi_2$ and $\pi_3$, we know that $\pi_1$ is parallel to $\pi_2$ and that $\pi_3$ intersects $\pi_1$ forming a line $r$.
- a) What will be the relative position of $\pi_3$ with respect to $\pi_2$?
- b) If we call $s$ the line of intersection between $\pi_3$ and $\pi_2$, what relationship of parallelism or perpendicularity do lines $r$ and $s$ have?
- c) If a line $t$ is perpendicular to $\pi_1$, will it also be perpendicular to $\pi_2$? Why?
Dihedral angles
A flat plane splits space in two half-planes when you pick a line in it. Two half-planes that share a line (the edge) form a dihedral angle, like an open book. The half-planes are its faces.
To measure a dihedral angle, slice it with a plane perpendicular to the edge. The two rays you get (one in each face) form an ordinary angle, called the plane angle, and its measure is the measure of the dihedral angle. A book lying flat is $180^\circ$. A half-open book with a $90^\circ$ opening has perpendicular covers. Two planes are perpendicular exactly when they form a $90^\circ$ dihedral angle.
Three half-planes meeting at one point form a trihedral angle. The corner of a room is one. Its three flat angles at the vertex (some problems call them rectilinear angles) are the angles between the pairs of edges.
When you open a math book 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…
When you open a math book 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 does this dihedral angle measure?
- b) If the book is half-open and the pages form an angle of $90^\circ$, what are the planes of the two pages like with respect to each other?
A revolving door in a shopping center has three identical glass leaves that intersect at a central vertical axis. a) What name is given to each of the three regions of…
A revolving door in a shopping center has three identical glass leaves that intersect at a central vertical axis.
- a) What name is given to each of the three regions of space bounded by two consecutive leaves?
- b) If the dihedral angle between the first and second leaf is $120^\circ$, and the three leaves are distributed symmetrically, what do the other two dihedral angles measure?
- c) If a fly lands exactly on the central axis, how many faces of the dihedral angles does it belong to?
Look at the corner of a room where two walls and the floor meet. This set forms a trihedral angle at the vertex of the bottom corner. a) What do the three rectilinear…
Look at the corner of a room where two walls and the floor meet. This set forms a trihedral angle at the vertex of the bottom corner.
- a) What do the three rectilinear angles that define the faces of this trihedron measure?
- b) What is the sum of the measures of the three dihedral angles formed between the walls and the floor?
- c) If a fly flies following a path parallel to the baseboard line of one wall, is that path also parallel to the plane of the other wall? Explain why.
Parallel and perpendicular in solids
Solids are made of planes and lines, so everything above applies to their faces and edges. Rotate this triangular prism and look for the facts below.
A plane of symmetry of a solid cuts it into two halves that are mirror images: reflecting the solid across that plane gives back the same solid. Counting planes of symmetry is a way to use the perpendicular and parallel facts above.
Consider a right prism with a regular hexagonal base. Analyze 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. Analyze 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?
Analyze a regular hexagonal prism. Consider a line $r$ passing through two opposite vertices of the bottom base and a line $s$ passing through two opposite vertices of…
Analyze a regular hexagonal prism. Consider a line $r$ passing through two opposite vertices of the bottom base and a line $s$ passing through two opposite vertices of the top base, but in such a way that they are not parallel to each other.
- a) Are lines $r$ and $s$ coplanar?
- b) Find the number of planes of symmetry of the prism that are perpendicular to the line joining the centers of both bases.
- c) If we draw a plane containing $r$ and perpendicular to the lateral faces it crosses, what geometric figure is obtained in the cross-section?
Imagine a regular tetrahedron resting on a table (horizontal plane). a) How many of its edges are perpendicular to each other? Justify your answer by analyzing the…
Imagine a regular tetrahedron resting on a table (horizontal plane).
- a) How many of its edges are perpendicular to each other? Justify your answer by analyzing the angles formed by the edges that do not share a vertex.
- b) If we draw a plane parallel to the base passing through the centroid of the tetrahedron, what is the relative position of this plane with respect to the four heights of the solid?
Challenge: skew lines and parallel planes
For two lines that are not parallel, there is always a plane that contains one line and is parallel to the other. Why? Through a point of line $a$, draw a line $b'$ parallel to $b$. Lines $a$ and $b'$ meet, so they determine a plane, and $b$ is parallel to it. This is the plane to use when a problem says "a plane containing $s$ and parallel to $r$".
Two ways to settle whether two lines (or cables, or edges) touch:
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Parallel planes. If the lines lie in two parallel planes, they are parallel or skew, never intersecting. Check the directions.
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Coordinates. Put one vertex of the box at the origin. Write each line as a moving point, and see if the two moving points can ever coincide.
In an exhibition hall in the shape of a rectangular prism with dimensions $10 \times 6 \times 4$ feet, a taut cable is installed joining a vertex on the floor with the…
In an exhibition hall in the shape of a rectangular prism with dimensions $10 \times 6 \times 4$ feet, a taut cable is installed joining a vertex on the floor with the midpoint of the opposite top edge. A second cable joins the center of the ceiling with the midpoint of one of the vertical edges.
- a) Determine if there is any point where the cables touch (intersecting lines) or if, on the contrary, they cross in space without touching.
- b) Is there any plane that is parallel to both cables simultaneously? Justify your answer based on the directions of the cables.
In an avant-garde architectural design, a building has been designed in the shape of a regular octahedron. If we draw a straight line $r$ that joins the centers of two…
In an avant-garde architectural design, a building has been designed in the shape of a regular octahedron. If we draw a straight line $r$ that joins the centers of two opposite faces and another line $s$ that contains one of the edges not belonging to either of those two faces, determine with reasoning:
- a) What is the relative position between line $r$ and line $s$?
- b) If we introduce a plane $\pi$ that contains the edge $s$ and is parallel to line $r$, what type of intersection would this plane have with the octahedron?
Challenge: planes perpendicular to a diagonal
How can you prove that a line is perpendicular to a plane when no right angle is drawn? Use the points that are the same distance from others.
The second fact follows from the first. $A$ and $G$ are both in the perpendicular bisecting plane of $BD$, so $AG \perp BD$. They are also both in the perpendicular bisecting plane of $BE$, so $AG \perp BE$. Then $AG$ is perpendicular to two intersecting lines of plane $BDE$, so it is perpendicular to the plane.
Consider a cube of edge $a$. A plane $\pi$ is drawn passing through the midpoints of three edges that meet at the same vertex. a) Justify whether the plane $\pi$ is…
Consider a cube of edge $a$. A plane $\pi$ is drawn passing through the midpoints of three edges that meet at the same vertex.
- a) Justify whether the plane $\pi$ is perpendicular to the diagonal of the cube starting from that vertex.
- b) If we repeat the process at the eight vertices of the cube, the eight planes cut the cube to form a new solid (a truncated cube). Analyze the relative position of the planes that have cut opposite vertices of the original cube.
Challenge: distances and cross-sections
The distance from a point to a line (or to a plane) is the length of the perpendicular segment from the point to it. That is the shortest path.
In a cone, a plane through the axis cuts a right triangle (the axial section) whose legs are the radius $R$ and the height $H$. Its hypotenuse is the slant height, the segment from the apex to the rim of the base. (Some problems call it a generator.) A shortcut for the perpendicular from the right angle to the hypotenuse is: $\text{altitude} = \dfrac{\text{leg}_1 \cdot \text{leg}_2}{\text{hypotenuse}}$.
For a sphere, a plane at distance $d$ from the center cuts a circle of radius $r$, where the radius of the sphere $R$ is the hypotenuse: $r = \sqrt{R^2 - d^2}$.
A point light source is placed at the center of the base of a right cone with radius $R = 5$ in and height $H = 12$ in. The source emits a ray of light that is…
A point light source is placed at the center of the base of a right cone with radius $R = 5$ in and height $H = 12$ in. The source emits a ray of light that is perpendicular to one of the generators of the cone.
- a) Is it possible for this ray to pass through the lateral surface of the cone?
- b) Calculate the distance from the light source to the point on the generator where the ray would be perpendicular. To do this, visualize the right-angled triangle formed by the axial section of the cone.
There is a sphere of radius $13$ in. A plane $\pi_1$ cuts the sphere at a distance of $5$ in from its center, generating a circle $C_1$. A second plane $\pi_2$,…
There is a sphere of radius $13$ in. A plane $\pi_1$ cuts the sphere at a distance of $5$ in from its center, generating a circle $C_1$. A second plane $\pi_2$, perpendicular to the first, also cuts 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 edge of the circle $C_1$ and the plane $\pi_2$? Use a drawing of the circular cross-section to reason your answer.
Back to the start: in your room, the floor and the ceiling are parallel planes, two walls are intersecting planes that meet along a corner (a line), and the desk edge and the vertical corner are skew lines, perpendicular skew lines if the desk is square with the wall.