How to form the transformation matrix for this

[Byang]

We were asked to form the transformation matrix that rotates the x₁ axis of a rectangular coordinate system 60 degrees toward x₂ and the x₃ axis.
The thing is, I don't understand what it meant by rotating one axis toward the two other. Like, do I rotate x₁ 60 degrees toward the x₂-x₃ plane or does it mean something else? I tried to rotate x₁ to the x₂-x₃ plane but I can never be sure of the angle between x₁' and x₂, and x₁' and x₃. How do I find the angles?

 

[Ray Vickson]

We were asked to form the transformation matrix that rotates the x₁ axis of a rectangular coordinate system 60 degrees toward x₂ and the x₃ axis.
The thing is, I don't understand what it meant by rotating one axis toward the two other. Like, do I rotate x₁ 60 degrees toward the x₂-x₃ plane or does it mean something else? I tried to rotate x₁ to the x₂-x₃ plane but I can never be sure of the angle between x₁' and x₂, and x₁' and x₃. How do I find the angles?

There are infinitely many possible ways of rotating the $x_1$ axis by $60^o$ towards the $x_2-x_3$ plane. For example, you could fix $x_2 = 0$ and rotate the $x_1$ axis towards the $x_3$ axis, or fix $x_3 = 0$ and rotate $x_1$ towards $x_2$, etc. In all these cases you would be rotating $x_1$ by $60^o$ towards the $x_2-x_3$ plane. In general, you could fix an arbitrary point $p = (0,p_2,p_3)$ in the $x_2-x_3$ plane, and rotate the $x_1$ axis towards $p$.

Basically, I don't know exactly what your question wants; perhaps they mean $60^o$ towards $x_2$ and then $60^o$ towards $x_3$, or maybe they mean two separate questions---one for an $x_1 \to x_2$ rotation and another for an $x_1 \to x_3$ rotation.

 

[Byang]

Basically, I don't know exactly what your question wants

Haha, that's exactly my problem. But thank you for answering. We've cleared it up with our teacher and she said it was supposed to be "around x₃ axis," which makes the problem a lot easier.

 

[Mark44]

Thread moved as it seems to be more of a general question than a specific homework problem.