Planes in Space: Quick Revision

A brief note on Planes in Space. Read it before a test. Try each question at the end before you open its answer.

The equation

• Vector form: $\vec{r} \cdot \vec{n} = \vec{a} \cdot \vec{n}$, with $\vec{n}$ a normal and $\vec{a}$ a point.
• Cartesian form: $Ax + By + Cz = k$, with normal $(A, B, C)$.
• Normal form: $\vec{r} \cdot \hat{n} = d$, and $d$ is the distance from the origin.
• Three points: a normal is $\overrightarrow{PQ} \times \overrightarrow{PR}$.
• Intercepts: $\dfrac{x}{a} + \dfrac{y}{b} + \dfrac{z}{c} = 1$.

Angles and the line of two planes

• Two 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 $0$.
• A line and a plane: $\sin\varphi = \dfrac{\lvert \vec{b} \cdot \vec{n} \rvert}{\lvert \vec{b} \rvert \lvert \vec{n} \rvert}$. A sine, not a cosine.
• Parallel line and plane: $\vec{b} \cdot \vec{n} = 0$.
• Where two planes meet: direction $\vec{n}_1 \times \vec{n}_2$, and a point from setting one coordinate to $0$.

Distance

• Point to plane: $\dfrac{\lvert Ax_1 + By_1 + Cz_1 + D \rvert}{\sqrt{A^2 + B^2 + C^2}}$.
• Parallel planes: $\dfrac{\lvert D_1 - D_2 \rvert}{\sqrt{A^2 + B^2 + C^2}}$, with equal coefficients.

A worked example

Find the plane through $P(1, 0, 0)$, $Q(0, 2, 1)$ and $R(2, 1, 3)$, and the distance of $(3, 1, 1)$ from it.

1. Two vectors. $\overrightarrow{PQ} = (-1, 2, 1)$ and $\overrightarrow{PR} = (1, 1, 3)$.
2. A normal. $\overrightarrow{PQ} \times \overrightarrow{PR} = (5, 4, -3)$.
3. The plane. $5x + 4y - 3z = 5$.
4. The distance. At $(3, 1, 1)$ the value is $15 + 4 - 3 - 5 = 11$. The length of the normal is $\sqrt{50}$. So $d = \dfrac{11}{\sqrt{50}} \approx 1.556$.

Practice with new numbers

A program to try

The program finds the plane of the worked example and the distance of the point from it.

Mistakes to avoid

• Using the position vectors of the points as the normal. A normal comes from a cross product.
• A cosine for a line and a plane. The formula gives the sine.
• Forgetting the modulus. Angles and distances are positive.
• Not dividing by the length of the normal. The value of $Ax + By + Cz + D$ is not a distance.

Check yourself

1. Find the plane through $(1, 2, 3)$ with normal $(2, 1, 2)$.

2. Find the distance of the origin from $2x + y + 2z = 10$.

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

4. A line has direction $(1, 2, 2)$ and a plane has normal $(2, 1, 0)$. Find $\sin\varphi$.

5. How do you find the direction of the line where two planes meet?

Sources

• OpenStax, Calculus Volume 3, Chapter 2, section on equations of lines and planes in space.
• NCERT, Mathematics Part II for Class XII, Chapter 11, section on the plane.