<p><strong>How to Prove the Formula of Distance between Two Parallel Planes</strong></p>

**How to Prove the Formula of Distance between Two Parallel Planes**

Vectors - General

Vectors

International Baccalaureate Higher Level Mathematics

IB HL Math

How to Prove the Formula of Distance between Two Parallel Planes

Let \(\pi_1: \mathbf{r} \cdot \mathbf{n} = k_1\) and \(\pi_2: \mathbf{r} \cdot \mathbf{n} = k_2\)

Let \(A\) and \(B\) be points on \(\pi_1\) and \(\pi_2\) respectively,
where \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\)

\(\Rightarrow \mathbf{a} \cdot \mathbf{n} = k_1\) (\(A\) satisfies \(\pi_1\)) and \(\mathbf{b} \cdot \mathbf{n} = k_2\) (\(B\) satisfies \(\pi_2\))

Distance between two parallel planes
\(= \left| \overrightarrow{AB} \cdot \hat{\mathbf{n}} \right|\)
\(= \left| (\overrightarrow{OB} - \overrightarrow{OA}) \cdot \displaystyle\frac{\mathbf{n}}{|\mathbf{n}|} \right|\)
\(= \displaystyle\frac{|(\mathbf{b} - \mathbf{a}) \cdot \mathbf{n}|}{|\mathbf{n}|}\)
\(= \displaystyle\frac{|\mathbf{b} \cdot \mathbf{n} - \mathbf{a} \cdot \mathbf{n}|}{|\mathbf{n}|}\)
\(= \displaystyle\frac{|k_2 - k_1|}{|\mathbf{n}|}\)

E.g. \(\pi_1: \mathbf{r} \cdot \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} = 4\), \(\pi_2: \mathbf{r} \cdot \begin{pmatrix} -1 \\ -2 \\ -1 \end{pmatrix} = 10\)

\(\pi_2: \mathbf{r} \cdot \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} = -10\)

Distance between two planes \(= \displaystyle\frac{|-10 - 4|}{\left| \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} \right|} = \frac{14}{\sqrt{1+4+1}} = \frac{14}{\sqrt{6}}\) units

How to Prove the Formula of Distance between Two Parallel Planes

How to Prove the Formula of Distance between Two Parallel Planes

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