What Are the Correct Euler's Angles for Rotating a Cube?

[Cosmossos]

Homework Statement

What are the Euler's angles corresponding to the rotations of a cube in $$\theta$$ radians around each of its principal axes



around the x: $$\theta$$=$$\theta$$
$$\phi$$=$$\psi$$=0

around z:$$\psi$$=$$\theta$$=0
$$\phi$$=$$\theta$$

around y:$$\phi$$=$$\theta$$=$$\theta$$
$$\psi$$=-$$\theta$$


is it correct? how can I make it more clear? It's very confusing...

 

[Cosmossos]

Help me please...

 

[Maxim Zh]

In order to formalize the solution you can use the rotation matrices: $$\hat{R}_x(\alpha)$$, $$\hat{R}_y(\alpha)$$, $$\hat{R}_z(\alpha)$$ and the matrix which describes the whole Euler's transform:

$$\hat{R}(\theta, \phi, \psi) = \hat{R}_z(\phi) \hat{R}_x(\theta) \hat{R}_z(\psi) \quad (1)$$

It's easy to get rotation around the x and z axes from (1) and your answers for these cases are right.

As for y-axis the condition

$$\hat{R}(\theta, \phi, \psi) = \hat{R}_y(\alpha)$$

yields

$$\theta = -\frac{\pi}{2} - \alpha;$$

$$\phi = \psi = -\frac{\pi}{2}.$$