Q 16. - Calculus

Question

Explain how to tell when two planes are perpendicular.

Step-by-Step Solution

Answer

The planes determined by the equations $a x+b y+c z=d$ and $\alpha x+\beta y+\gamma z=\delta$ are perpendicular if and only if $a\alpha +b\beta +c\gamma =0$

Step 1: Given information

The planes determined by the equations $a x+b y+c z=d$ and $\alpha x+\beta y+\gamma z=\delta$

Step 2: Calculation

The goal is to demonstrate how to detect if two planes are perpendicular to one another.

The normal vector of the equation $a x+b y+c z=d$ is ${\mathbf{N}}_1=⟨a,b,c⟩$ and of the equation $\alpha x+\beta y+\gamma z=\delta$ is ${\mathbf{N}}_2=⟨\alpha ,\beta ,\gamma ⟩$

Step 3: Calculation

Only if and only if the planes' normal vectors are orthogonal are they considered orthogonal.

The normal vectors are orthogonal if and only if ${\mathbf{N}}_1·{\mathbf{N}}_2=0$

The orthogonal condition ${\mathbf{N}}_1·{\mathbf{N}}_2=0$ gives:

$$\begin{gathered} ⟨a,b,c⟩·⟨\alpha ,\beta ,\gamma ⟩=0 \\ a\alpha +b\beta +c\gamma =0\text{ (Dot Product) } \end{gathered}$$

Thus, the planes determined by the equations $a x+b y+c z=d$ and $\alpha x+\beta y+\gamma z=\delta$ are perpendicular if and only if $a\alpha +b\beta +c\gamma =0$