The Angle Between Two Planes

The angle between two planes

Two planes that are not parallel meet along a line, like the two covers of a half-open book meeting along the spine. The angle between the planes is the angle you would measure across that crease, in a slice square to it, like the opening of the book.

Look along the crease, so that each plane is seen edge-on as a line. Each normal is then its plane’s line turned a quarter turn, 90°. Turning both lines by the same quarter turn does not change the angle between them. So the angle between the planes is the angle between their normals: $\cos \theta =n_1\cdot n_2/\left(|n_1||n_2|\right)$cos θ = n₁ · n₂ / (|n₁| |n₂|).

This is why the normals are the right vectors to use. Directions lying inside the planes can point any way within them, and the crease lies in both planes at once, so neither of those fixes the angle.

Two planes seen edge-on, looking along their crease: plane 1 is level and plane 2 rises 3 for every 4 across, so they meet at 36.9°. The normals $n_1=\left(0,2.5\right)$n₁ = (0, 2.5) and $n_2=\left(-1.5,2\right)$n₂ = (−1.5, 2) meet at the same angle, since $n_1\cdot n_2/\left(|n_1||n_2|\right)=5/6.25=0.8$n₁ · n₂ / (|n₁| |n₂|) = 5 / 6.25 = 0.8 and ${\cos }^{-1}0.8=36.9^{\circ }$cos⁻¹ 0.8 = 36.9°.

A plane with its normal of length 1, here n = (0.57, 0, 0.82) to two decimal places. Against the level normal (0, 0, 1), $\cos \theta =0.82$cos θ = 0.82, so n leans 35° from straight up, and the plane leans the same 35° from level. Tilt n and the plane tips by exactly the angle n turns.

Reading the normals

Take the planes 2x + y + 2z = 5 and x + 2y + 2z = 1. The coefficients are the normals: $n_1=\left(2,1,2\right)$n₁ = (2, 1, 2) and $n_2=\left(1,2,2\right)$n₂ = (1, 2, 2).

Their lengths are $|n_1|=\sqrt{4+1+4}=3$|n₁| = √(4 + 1 + 4) = 3 and $|n_2|=\sqrt{1+4+4}=3$|n₂| = √(1 + 4 + 4) = 3. The constants 5 and 1 say where each plane is; they play no part in the angle.

The dot product and the angle

The dot product is $n_1\cdot n_2=2\times 1+1\times 2+2\times 2=2+2+4=8$n₁ · n₂ = 2 × 1 + 1 × 2 + 2 × 2 = 2 + 2 + 4 = 8.

Divide by both lengths: $\cos \theta =8/\left(3\times 3\right)=\frac{8}{9}$cos θ = 8 / (3 × 3) = 8/9, so $\theta ={\cos }^{-1}\left(\frac{8}{9}\right)=27.3^{\circ }$θ = cos⁻¹(8/9) = 27.3° to one decimal place. The two planes are close to parallel, tilted only 27.3° apart.

A second pair

The planes x + y = 3 and y + z = 5 have normals (1, 1, 0) and (0, 1, 1). The dot product is 0 + 1 + 0 = 1, and each normal has length $\sqrt{2}$√2, so $\cos \theta =1/\left(\sqrt{2}\times \sqrt{2}\right)=\frac{1}{2}$cos θ = 1 / (√2 × √2) = 1/2 and $\theta =60^{\circ }$θ = 60°.

Parallel planes, and planes at right angles

If one normal is a multiple of the other, the normals point along one line, $\theta =0^{\circ }$θ = 0°, and the planes are parallel. The planes x + 2y + 2z = 3 and 2x + 4y + 4z = 5 have normals (1, 2, 2) and (2, 4, 4) = 2(1, 2, 2), so they are parallel.

Whether two such planes are the same plane is a separate question, and the constants answer it. Doubling x + 2y + 2z = 3 gives 2x + 4y + 4z = 6, so 2x + 4y + 4z = 6 is the same plane and 2x + 4y + 4z = 5 is a different, parallel one. Equal constants do not decide it: 2x + 4y + 4z = 3 is also parallel and different.

If $n_1\cdot n_2=0$n₁ · n₂ = 0, $\cos \theta =0$cos θ = 0 and $\theta =90^{\circ }$θ = 90°: the planes meet at right angles, like a wall and the floor. The planes 2x + y + 2z = 5 and x − z = 4 have normals (2, 1, 2) and (1, 0, −1), and 2 + 0 − 2 = 0.

Two angles at every crease

Two crossing planes make two angles across the crease, and they add to 180°. Which one the formula gives depends on which way each normal points. Replace $n_2$n₂ by $-n_2=\left(-1,-2,-2\right)$−n₂ = (−1, −2, −2), which is just as good a normal for the same plane, and the dot product becomes −8, so $\cos \theta =-\frac{8}{9}$cos θ = −8/9 and $\theta =152.7^{\circ }$θ = 152.7°.

Both 27.3° and 152.7° are angles between the planes. The angle between two planes is normally given as the acute one, so take the size of the cosine: $\cos \theta =|n_1\cdot n_2|÷\left(|n_1||n_2|\right)=\frac{8}{9}$cos θ = |n₁ · n₂| ÷ (|n₁| |n₂|) = 8/9, and $\theta =27.3^{\circ }$θ = 27.3°.

The edge-on pair again with $n_2$n₂ reversed. The normals now meet at 180° − 36.9° = 143.1°, the obtuse angle between the planes; the acute angle, 36.9°, is the one quoted.

The usual mistakes

Using directions inside the planes, or the crease itself. Only the normals fix how each plane faces.

Dividing by one length. The normals (3, 0, 4) and (4, 0, 3) have dot product 24 and lengths 5 and 5, so $\cos \theta =24/25$cos θ = 24 / 25. Stopping at 24 ÷ 5 gives more than 1, which no cosine can be.

Using the raw dot product. 24 is not a cosine; it grows with the lengths of the normals.

Letting the constants decide whether planes are parallel. Only the normals decide that; the constants then say whether two parallel planes are one plane or two.

A panel on a roof

In the application below, a roof facing east and a solar panel turned to face south are two planes. Each normal is read off its equation, and the dot product gives the angle between them, and then the panel’s tilt from the horizontal.