Angles: plane with plane, and line with plane

What you will be able to do

• find the angle between two planes, and test whether planes are parallel or perpendicular;
• find the angle between a line and a plane, and test whether a line is parallel to a plane or lies in it;
• find the point where a line meets a plane.

Two roofs meeting at a ridge

Two sloping roofs meet along a ridge. The angle between the roofs is the angle you would measure in a cross-section, at right angles to the ridge. It equals the angle between the two normals, because turning a plane turns its normal by the same angle.

The angle between two planes

For planes $\vec{r} \cdot \vec{n}_1 = k_1$ and $\vec{r} \cdot \vec{n}_2 = k_2$, the angle $\theta$ between them satisfies

In coordinates, the dot product is $A_1A_2 + B_1B_2 + C_1C_2$, and each length is the square root of the sum of the squares of the three coefficients.

• Perpendicular planes: $A_1A_2 + B_1B_2 + C_1C_2 = 0$.
• Parallel planes: the normals are proportional, $\dfrac{A_1}{A_2} = \dfrac{B_1}{B_2} = \dfrac{C_1}{C_2}$.

Two planes make two angles, $\theta$ and $180^\circ - \theta$. As for lines, we report the acute one.

The angle between a line and a plane

A line with direction $\vec{b}$ meets a plane with normal $\vec{n}$. The angle $\varphi$ between the line and the plane is measured from the plane, so it is $90^\circ$ minus the angle between $\vec{b}$ and $\vec{n}$. Since $\cos(90^\circ - \varphi) = \sin\varphi$,

This is a sine, not a cosine. The most common mistake is to treat it like the angle between two lines.

• The line is parallel to the plane: $\vec{b} \cdot \vec{n} = 0$.
• The line is perpendicular to the plane: $\vec{b}$ is parallel to $\vec{n}$.
• The line lies in the plane: $\vec{b} \cdot \vec{n} = 0$, and one point of the line is on the plane.

Where a line meets a plane

Write a point of the line as $\vec{a} + \lambda\vec{b}$ and put it in the equation of the plane. This gives one equation in $\lambda$. If $\vec{b} \cdot \vec{n} \ne 0$ there is exactly one solution, and it fixes the point.

A worked example

Two planes. Find the angle between $x + 2y + 2z = 7$ and $2x - 3y + 6z = 12$. The normals are $(1, 2, 2)$ and $(2, -3, 6)$. The dot product is $2 - 6 + 12 = 8$, and the lengths are $3$ and $7$. So $\cos\theta = \dfrac{8}{21}$ and $\theta \approx 67.61^\circ$.

A line and a plane. The line through $(1, -1, 2)$ with direction $(2, 3, 6)$ meets the plane $x + 2y + 2z = 23$.

1. The point. $(1 + 2\lambda) + 2(-1 + 3\lambda) + 2(2 + 6\lambda) = 3 + 20\lambda = 23$, so $\lambda = 1$ and the point is $(3, 2, 8)$.
2. The angle. $\vec{b} \cdot \vec{n} = 2 + 6 + 12 = 20$. The lengths are $7$ and $3$. So $\sin\varphi = \dfrac{20}{21}$ and $\varphi \approx 72.25^\circ$.

A parallel case. The line with direction $(2, 1, -2)$ has $\vec{b} \cdot \vec{n} = 2 + 2 - 4 = 0$ with the normal $(1, 2, 2)$. It is parallel to the plane.

Practise with new numbers

Try it in Python

The program finds the angle between two planes and the angle between a line and a plane.

Common mistakes

• Using a cosine for a line and a plane. The formula with $\vec{b} \cdot \vec{n}$ gives $\sin\varphi$.
• Dropping the modulus. The angle between two planes, or a line and a plane, is acute.
• Using a point of the plane as a direction. The normal comes from the coefficients $(A, B, C)$.
• Forgetting to test the point. A line with $\vec{b} \cdot \vec{n} = 0$ is parallel to the plane, and it lies in the plane only if one point of the line is also on it.

Remember these

• Planes: $\cos\theta = \dfrac{\lvert \vec{n}_1 \cdot \vec{n}_2 \rvert}{\lvert \vec{n}_1 \rvert \lvert \vec{n}_2 \rvert}$. Perpendicular means a dot product of $0$.
• Line and plane: $\sin\varphi = \dfrac{\lvert \vec{b} \cdot \vec{n} \rvert}{\lvert \vec{b} \rvert \lvert \vec{n} \rvert}$.
• A line is parallel to a plane when $\vec{b} \cdot \vec{n} = 0$.
• To meet a line and a plane, put $\vec{a} + \lambda\vec{b}$ in the equation of the plane.

Check yourself

Try each one on your own first. If you are stuck, open a hint. Then check with Show the answer.

1. Are the planes $2x - y + 2z = 5$ and $x + 2y = 1$ perpendicular?

2. Find the angle between the planes $x + y = 3$ and $y + z = 5$.

3. A line with direction $(1, 1, 1)$ meets the plane $x - y + 2z = 4$. Find $\sin\varphi$.

4. Where does the line through the origin with direction $(1, 2, 3)$ meet the plane $x + y + z = 12$?

5. A line has direction $(3, 0, -2)$ and passes through $(1, 1, 1)$. Does it lie in the plane $2x + y + 3z = 6$?

References

• OpenStax, Calculus Volume 3, Chapter 2, sections on the dot product and on equations of lines and planes in space (angle between planes).
• NCERT, Mathematics Part II for Class XII, Chapter 11, section on the angle between two planes and between a line and a plane.