A unit vector $\vec{a}$ is such that it makes an angle $\frac{\pi}{4}$ with the $x$-axis, $\frac{\pi}{3}$ with the $y$-axis and an acute angle $\theta$ with the $z$-axis. Find $\theta$ and the components of $\vec{a}$.
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Remember the classic standard values for direction cosines: $\cos^2(45^\circ) + \cos^2(60^\circ) + \cos^2(60^\circ) = \frac{1}{2} + \frac{1}{4} + \frac{1}{4} = 1$. Recognizing these standard pythagorean relations can save you valuable calculation time during exams.