Solution: The magnitude of the cross product is $\|\mathbf{p} \times \mathbf{q}\| = \|\mathbf{p}\| \|\mathbf{q}\| \sin\theta$. Since $\mathbf{p}$ and $\mathbf{q}$ are unit vectors, $\|\mathbf{p} \times \mathbf{q}\| = \sin\theta = \frac{1}{2}$. Thus, $\theta = \arcsin\left(\frac{1}{2}\right)$, which gives $\theta = \frac{\pi}{6}$ or $\frac{5\pi}{6}$. However, the angle between vectors is taken as the acute angle, so $\theta = \boxed{\dfrac{\pi}{6}}$.

["Understanding the Cross Product Magnitude: The Key Role of Unit Vectors", "The cross product of two vectors, frequently encountered in physics and engineering, provides both magnitude and direction, making it essential for calculating areas, detecting perpendicularity, and describing rotational effects. A fundamental formula governing the cross product’s magnitude is:", "[\n|\mathbf{p} \ imes \mathbf{q}| = |\mathbf{p}| |\mathbf{q}| \sin\ heta\n]", "where $\ heta$ is the angle between vectors $\mathbf{p}$ and $\mathbf{q}$. This elegant equation reveals that the strength of the cross product depends not only on the lengths of the vectors but also on the sine of the angle between them.", "### Why Unit Vectors Matter", "Consider the special case where both $\mathbf{p}$ and $\mathbf{q}$ are unit vectors—meaning $|\mathbf{p}| = 1$ and $|\mathbf{q}| = 1$. The formula simplifies to:", "[\n|\mathbf{p} \ imes \mathbf{q}| = \sin\ heta\n]", "Given that $|\mathbf{p} \ imes \mathbf{q}| = \frac{1}{2}$, we immediately find:", "[\n\sin\ heta = \frac{1}{2}\n]", "Solving for $\ heta$, we know $\ heta = \arcsin\left(\frac{1}{2}\right)$, which yields two possible solutions:", "[\n\ heta = \frac{\pi}{6} \quad \ ext{or} \quad \ heta = \frac{5\pi}{6}\n]", "However, in vector geometry, the angle between two vectors is conventionally defined as the acute angle—between $0$ and $\pi/2$. Therefore, while $\frac{5\pi}{6}$ is a valid solution mathematically, the physically meaningful and standard interpretation takes the smaller acute angle.", "Thus, the magnitude of the cross product corresponds to:", "[\n|\mathbf{p} \ imes \mathbf{q}| = \frac{1}{2} \implies \ heta = \boxed{\dfrac{\pi}{6}}\n]", "### Practical Implications", "This concept underpins many applications: from torque calculations in mechanics to electromagnetic force directions in physics. Knowing that unit vectors reduce the problem to a simple sine function simplifies analysis and ensures consistency across computations.", "### Summary", "When working with cross products and unit vectors, always remember:", "- The magnitude of the cross product equals the product of the magnitudes times the sine of the angle between them.\n- For unit vectors, $|\mathbf{p} \ imes \mathbf{q}| = \sin\ heta$.\n- Using $\sin\ heta = \frac{1}{2}$ gives two possible angles, but only $\frac{\pi}{6}$ is taken as the standard angle between vectors.", "This insight streamlines both theoretical understanding and practical problem-solving involving vector cross products."]