Grade 11

Parallelism and perpendicularity in space

How lines and planes relate in three dimensions: parallel, intersecting or perpendicular.

IntuitionA room full of parallel and perpendicular lines

Look around any room: the wall meets the floor along a straight edge, opposite walls never meet however far they extend, and a plumb line hanging from the ceiling drops straight down, perpendicular to the floor. Solid geometry gives precise names to these everyday relationships between lines and planes: parallel (never meeting, same "direction"), intersecting (crossing along a line), and perpendicular (meeting at a right angle). The interactive cube below lets you see all three at once.

A cube: opposite faces lie in parallel planes, edges meeting at a vertex are perpendicular, and the two face diagonals on opposite faces (for example the diagonal of the bottom face and the diagonal of the top face going the other way) are skew: neither parallel nor intersecting.

SchoolParallelism between lines and planes

Definition: Line parallel to a plane; two parallel planes

A line $d$ is parallel to a plane $(\alpha)$ if they have no common point. Two planes $(\alpha)$ and $(\beta)$ are parallel if they have no common point (the case where they coincide is excluded; they must be genuinely disjoint).

d \parallel a,\quad a \subset (\alpha),\quad d \not\subset (\alpha) \ \implies\ d \parallel (\alpha)

This is the standard test for a line being parallel to a plane: if $d \parallel a$ for some line $a$ already lying in the plane ( $a \subset (\alpha)$ ), and $d$ itself is not in the plane ( $d \not\subset (\alpha)$ ), then $d \parallel (\alpha)$ . In words: to show a line avoids an entire plane, it is enough to find one line inside the plane that it is parallel to.

(\alpha) \parallel (\beta) \iff \exists\, a, b \subset (\alpha),\ a \cap b = \{O\},\ a \parallel (\beta),\ b \parallel (\beta)

Two planes are parallel exactly when we can find two intersecting lines $a, b \subset (\alpha)$ ( $a \cap b = \{O\}$ , so they are not parallel to each other) that are each parallel to the other plane ( $a \parallel (\beta)$ and $b \parallel (\beta)$ ). One line alone is not enough — a single line parallel to $(\beta)$ could still let $(\alpha)$ tilt and cut through $(\beta)$ ; a second, intersecting line pins the whole plane down.

Relations between lines and planes in space
Relation	Defining condition
Two lines parallel	Coplanar, no common point
Two lines skew	Not coplanar (no common plane)
Line parallel to a plane	$d \parallel (\alpha)$ : no common point, line not in the plane
Two planes parallel	$(\alpha) \parallel (\beta)$ : no common point
Line perpendicular to a plane	$d \perp (\alpha)$ : perpendicular to every line of the plane

UndergraduatePerpendicularity: rigorous definitions and theorems

Definition: Line perpendicular to a plane

A line $d$ is perpendicular to a plane $(\alpha)$ , written $d \perp (\alpha)$ , if it is perpendicular to every line contained in that plane, not just to one or two of them.

d \perp (\alpha) \iff \forall\, c \subset (\alpha),\ d \perp c

Theorem: Line–plane perpendicularity criterion

If $d \perp a$ and $d \perp b$ for two intersecting lines $a, b \subset (\alpha)$ with $a \cap b = \{O\}$ , then $d \perp (\alpha)$ .

Why is it true?

You do not need to check infinitely many lines in the plane. Just as two nails driven perpendicular to a floor along two different directions from the same point are enough to hold a post exactly upright, two independent perpendicularity checks pin down perpendicularity to the whole plane.

Proof

Place the intersection point $O$ at the origin, and let $\vec{u}$ and $\vec{v}$ be direction vectors of $a$ and $b$ . Since $a$ and $b$ intersect and are distinct lines, $\vec{u}$ and $\vec{v}$ are linearly independent, so together they span every direction lying in the plane $(\alpha)$ : the direction vector of any line $c \subset (\alpha)$ can be written as $\vec{w} = \lambda \vec{u} + \mu \vec{v}$ for some real numbers $\lambda, \mu$ .

Let $\vec{n}$ be a direction vector of $d$ . The hypotheses $d \perp a$ and $d \perp b$ translate to the dot-product equations $\vec{n} \cdot \vec{u} = 0$ and $\vec{n} \cdot \vec{v} = 0$ . For a line $c$ in $(\alpha)$ with direction $\vec{w} = \lambda \vec{u} + \mu \vec{v}$ , linearity of the dot product gives $\vec{n} \cdot \vec{w} = \lambda(\vec{n}\cdot\vec{u}) + \mu(\vec{n}\cdot\vec{v}) = \lambda \cdot 0 + \mu \cdot 0 = 0$ .

Since the dot product of $\vec{n}$ with the direction of an arbitrary line $c \subset (\alpha)$ is zero, $d$ is perpendicular to every line of $(\alpha)$ , which is exactly the definition of $d \perp (\alpha)$ .

Theorem: The Three Perpendiculars Theorem

Let a line $d$ not lie in and not be perpendicular to a plane $(\alpha)$ , and let $d_1$ be the orthogonal projection of $d$ onto that plane $(\alpha)$ . Then for any line $a$ contained in $(\alpha)$ : $a \perp d$ if and only if $a \perp d_1$ .

Why is it true?

Checking perpendicularity to a slanted (oblique) line directly is awkward, but its shadow $d_1$ lies flat inside the plane, where perpendicularity is easy to see and measure. This theorem says the two checks always agree, so you may always replace the hard 3D check with the easy 2D one.

Proof

Fix a point $A$ on $d$ outside $(\alpha)$ , and let $H$ be the foot of the perpendicular from $A$ to $(\alpha)$ , so $AH \perp (\alpha)$ ; by definition of orthogonal projection, $H$ lies on $d_1$ . Because $AH$ is perpendicular to the entire plane $(\alpha)$ , it is perpendicular to the line $a$ inside that plane: $AH \perp a$ . Meanwhile $d$ , $d_1$ and $AH$ all lie in one vertical plane.

( $\Rightarrow$ ) Suppose $a \perp d$ . Then $a$ is perpendicular to both $d$ and $AH$ , which are two intersecting lines of that vertical plane. By the line–plane perpendicularity criterion, $a$ is perpendicular to the whole vertical plane, and therefore to $d_1$ , which lies in it; hence $a \perp d_1$ .

( $\Leftarrow$ ) Conversely, suppose $a \perp d_1$ . Then $a$ is perpendicular to both $d_1$ and $AH$ , again two intersecting lines of the same vertical plane. The same criterion makes $a$ perpendicular to that vertical plane, and therefore to $d$ ; hence $a \perp d$ .

Definition: Dihedral angle and distance between skew lines

When two planes $(\alpha)$ and $(\beta)$ intersect along an edge $c$ , pick a point $I$ on $c$ and draw in each plane the line through $I$ perpendicular to $c$ : $p \subset (\alpha)$ with $p \perp c$ , and $q \subset (\beta)$ with $q \perp c$ . The angle $\big((\alpha), (\beta)\big) = (p, q)$ between $p$ and $q$ is the plane angle of the dihedral angle. For two skew lines $a$ and $b$ , there is a unique common perpendicular segment $MN$ ( $M$ on $a$ , $N$ on $b$ , $MN \perp a$ , $MN \perp b$ ); its length $d(a, b) = MN$ is the shortest distance between the two lines.

d(a, b) = \min_{P \in a,\, Q \in b} PQ = MN,\qquad MN \perp a,\ MN \perp b

UndergraduateReal-World Applications and Worked Examples

Every building column checked with a spirit level along two perpendicular walls uses the line–plane perpendicularity criterion; aircraft designers choose the upward "dihedral angle" between left and right wings so a rolling plane naturally rights itself; and roboticists and civil engineers compute the skew-line distance between two non-parallel pipes, cables or robot links to verify they will not collide.

Example: Roof pitch (dihedral angle) via the Three Perpendiculars Theorem

A pavilion roof has the shape of a pyramid $S.ABCD$ over a square floor $ABCD$ of side $4$ m, with the vertical post $SA = 4$ m standing perpendicular to the floor ( $SA \perp (ABCD)$ ). Find the dihedral angle that the sloped roof panel $(SBC)$ makes with the floor $ABCD$ along the eave $BC$ .

Solution

Because $SA \perp (ABCD)$ , the orthogonal projection of the slanted edge $SB$ onto the floor $ABCD$ is the side $AB$ . In the square $ABCD$ , the eave $BC$ is perpendicular to $AB$ ( $AB \perp BC$ ). By the Three Perpendiculars Theorem, $BC$ is also perpendicular to the slanted edge $SB$ ( $SB \perp BC$ ).

Since $AB$ (in the floor) and $SB$ (in the roof panel $(SBC)$ ) both meet the eave $BC$ at right angles at the same point, the dihedral angle along $BC$ is simply the acute angle $\angle SBA$ of the right triangle $\triangle SAB$ (right-angled at $A$ because $SA$ is vertical). Computing $\tan(\angle SBA) = \frac{SA}{AB} = \frac{4}{4} = 1$ gives $\angle SBA$ = $45^\circ$ .

Example: Clearance between two skew cables in a cubic frame

In a cubic steel frame $ABCD.A'B'C'D'$ of edge length $6$ m, one vertical conduit runs along the corner post $AA'$ while a diagonal tension cable runs across the floor along $BD$ . Find the shortest distance between these two skew lines.

Solution

Let $O$ be the center of the square floor, where the two floor diagonals meet. In a square, the diagonals bisect each other at right angles, so the half-diagonal $AO$ satisfies $AO \perp BD$ .

Meanwhile, the vertical corner post satisfies $AA' \perp (ABCD)$ , and since $AO$ lies in the floor plane, we also have $AO \perp AA'$ . Therefore $AO$ meets both $AA'$ (at $A$ ) and $BD$ (at $O$ ) at right angles — it is the unique common perpendicular segment of the two skew lines. Its length is half the diagonal of a square of side $6$ m: $AO = \frac{1}{2} AC = \frac{6\sqrt{2}}{2} = 3\sqrt{2}$ m.

To conclude $d \perp (\alpha)$ from $d \perp a$ and $d \perp b$ with $a$ , $b$ lying in $(\alpha)$ , what extra condition on $a$ and $b$ is required?

In the pyramid $S.ABCD$ with square base $ABCD$ , $SA \perp (ABCD)$ and $SA = AB = a$ . What is the dihedral angle between the lateral face $(SBC)$ and the base $ABCD$ ?

In a cube $ABCD.A'B'C'D'$ of edge length $6$ , what is the shortest distance between the vertical edge $AA'$ and the base diagonal $BD$ ?

A carpenter checks a vertical timber post with a spirit level along two chalk lines drawn on a flat concrete slab. How should the two chalk lines be chosen so that passing both checks guarantees the post is perpendicular to the slab?

References

Euclid; trans. T. L. Heath (1908). Euclid's Elements, Book XI (perpendicularity and parallelism of lines and planes)
Wikipedia contributors (2024). Dihedral angle