Question

Explain why two planes orthogonal to the same vector are either parallel or identical.

Short answer

The two planes orthogonal to the same vector are either parallel or identical.

Step-by-step solution

Given information

Consider the two planes with normal vectors ${\mathbf{N}}_1$ and ${\mathbf{N}}_2$ orthogonal to the same non-zero vector $v$

Calculation

The goal is to show that two orthogonal planes facing the same vector are either parallel or identical.

Use the fact that the dot product of orthogonal vectors is zero to verify the point.

The plane with normal vectors ${\mathbf{N}}_1$is orthogonal to the vector $\mathbf{v}$

Therefore,

${\mathbf{N}}_1·\mathbf{v}=0\text{......(1)}$

The plane with normal vectors ${\mathbf{N}}_2$ is orthogonal to the vector $\mathbf{v}$

Therefore,

${\mathbf{N}}_2·\mathbf{v}=0\dots \dots \left(2\right)$

Calculation

From equations $\left(1\right)$ and $\left(2\right)$ the following result is obtained.

$$\begin{gathered} {\mathbf{N}}_1·\mathbf{v}={\mathbf{N}}_2·\mathbf{v} \\ {\mathbf{N}}_1·\mathbf{v}-{\mathbf{N}}_2·\mathbf{v}=0 \\ \left({\mathbf{N}}_1-{\mathbf{N}}_2\right)·\mathbf{v}=0\text{(Dot product is associative)} \end{gathered}$$

Because vector vis is non-zero.

Therefore,

$$\begin{gathered} {\mathbf{N}}_1-{\mathbf{N}}_2=0 \\ {\mathbf{N}}_1={\mathbf{N}}_2 \end{gathered}$$

Therefore, the two planes orthogonal to the same vector are either parallel or identical.