Step 1: Recall the direction cosine identity.
If a ray makes angles $\alpha, \beta, \gamma$ with the x, y, z axes respectively, then $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$. Here $\alpha, \beta = 2\alpha, \gamma = 3\alpha$.
Step 2: Write the equation.
\[ \cos^2\alpha + \cos^2 2\alpha + \cos^2 3\alpha = 1. \]
Step 3: Use double and triple angle formulas.
Express everything in terms of $\cos 2\alpha$ and $\cos 6\alpha$: \[ \frac{1+\cos 2\alpha}{2} + \frac{1+\cos 4\alpha}{2} + \frac{1+\cos 6\alpha}{2} = 1. \] \[ 3 + \cos 2\alpha + \cos 4\alpha + \cos 6\alpha = 2. \] \[ \cos 2\alpha + \cos 4\alpha + \cos 6\alpha = -1. \]
Step 4: Group and simplify.
$\cos 2\alpha + \cos 6\alpha = 2\cos 4\alpha \cos 2\alpha$. So: $2\cos 4\alpha \cos 2\alpha + \cos 4\alpha = -1 \Rightarrow \cos 4\alpha(2\cos 2\alpha + 1) = -1$.
Step 5: Test $\alpha = \pi/6$.
$\cos^2(\pi/6) = 3/4$, $\cos^2(\pi/3) = 1/4$, $\cos^2(\pi/2) = 0$. Sum $= 3/4 + 1/4 + 0 = 1$. Yes!
Step 6: Test $\alpha = \pi/4$.
$\cos^2(\pi/4) = 1/2$, $\cos^2(\pi/2) = 0$, $\cos^2(3\pi/4) = 1/2$. Sum $= 1/2 + 0 + 1/2 = 1$. Yes! So both $\alpha = \pi/6$ and $\alpha = \pi/4$ work.
\[ \boxed{\alpha = \dfrac{\pi}{6},\ \dfrac{\pi}{4}} \]