Vectors

International Baccalaureate Higher Level Mathematics

IB HL Math

How to Prove the Formula of Distance between Two Parallel Planes

Let A and B be points on \(\Pi_1\) and \(\Pi_2\) respectively, where \(\overrightarrow{OA} = \mathbf{a}\) and \(\overrightarrow{OB} = \mathbf{b}\).

Since A is on \(\Pi_1\): \(\mathbf{a} \cdot \mathbf{n} = k_1\). Since B is on \(\Pi_2\): \(\mathbf{b} \cdot \mathbf{n} = k_2\).

\(= |\overrightarrow{AB} \cdot \hat{\mathbf{n}}| = \left|(\overrightarrow{OB} - \overrightarrow{OA}) \cdot \displaystyle\frac{\mathbf{n}}{|\mathbf{n}|}\right| = \left|\displaystyle\frac{(\mathbf{b} - \mathbf{a}) \cdot \mathbf{n}}{|\mathbf{n}|}\right|\)

\(= \displaystyle\frac{|\mathbf{b} \cdot \mathbf{n} - \mathbf{a} \cdot \mathbf{n}|}{|\mathbf{n}|} = \frac{|k_2 - k_1|}{|\mathbf{n}|}\)

Example: \(\Pi_1: \mathbf{r} \cdot \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} = 4\), \(\Pi_2: \mathbf{r} \cdot \begin{pmatrix} -1 \\ -2 \\ -1 \end{pmatrix} = 10\)

Rewriting \(\Pi_2\): \(\mathbf{r} \cdot \begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix} = -10\)

Distance \(= \displaystyle\frac{|-10 - 4|}{|\begin{pmatrix} 1 \\ 2 \\ 1 \end{pmatrix}|} = \frac{14}{\sqrt{6}}\) 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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