Q 14. - Calculus

Question

The angle$\theta$ between two intersecting planes called the dihedral angle is defined to be the angle between the two normal vectors to the planes, where

$\theta ={\cos }^{-1}\left(\frac{{\mathbf{N}}_1·{\mathbf{N}}_2}{||{\mathbf{N}}_1||||{\mathbf{N}}_2||}\right)$

Draw a figure that illustrates the dihedral angle and explain why the definition given is a reasonable definition.

Step-by-Step Solution

Answer

The dihedral angle definition is plausible.

Step 1: Given information

The normal vectors of two intersecting planes is ${\mathbf{N}}_1$ and ${\mathbf{N}}_2$

Step 2: Calculation

The normal vectors of two intersecting planes is ${\mathbf{N}}_1$ and ${\mathbf{N}}_2$

The angle between two planes is given by:

$$\begin{gathered} \cos \theta =\frac{{\mathbf{N}}_1·{\mathbf{N}}_2}{||{\mathbf{N}}_1||||{\mathbf{N}}_2||} \\ \theta ={\cos }^{-1}\left(\frac{{\mathbf{N}}_1·{\mathbf{N}}_2}{||{\mathbf{N}}_1||||{\mathbf{N}}_2||}\right) \end{gathered}$$

That's why the definition of the dihedral angle is reasonable.