The argument of $ z_1 \overline{z_2} $ is $ heta_1 - heta_2 $ (modulo $ 360^\circ $), and the smallest positive argument is given to be $ 60^\circ $. Since cosine determines the angle uniquely within $ [0^\circ, 180^\circ] $ for this context (as it's the smallest angle between them), we have

["The Argument of $ z_1 \overline{z_2} $: Understanding $ \arg(z_1 \overline{z_2}) = \ heta_1 - \ heta_2 \mod 360^\circ $, with Smallest Positive Argument at $ 60^\circ $", "When working with complex numbers in polar form, multiplying two complex numbers introduces a fundamental geometric principle: the argument (angle) of the product is the difference of their individual arguments. This concept becomes especially important when analyzing $ z_1 \overline{z_2} $, a quantity central in complex arithmetic with magnitudes but distinct directions.", "For two complex numbers\n$ z_1 = r_1 (\cos\ heta_1 + i\sin\ heta_1) $,\n$ z_2 = r_2 (\cos\ heta_2 + i\sin\ heta_2) $,", "the conjugate of $ z_2 $ is $ \overline{z_2} = r_2 (\cos\ heta_2 - i\sin\ heta_2) = r_2 (\cos(-\ heta_2) + i\sin(-\ heta_2)) $.\nMultiplying $ z_1 $ and $ \overline{z_2} $ gives:\n$$\nz_1 \overline{z_2} = r_1 r_2 \left[ (\cos\ heta_1 + i\sin\ heta_1)(cos(-\ heta_2) + i\sin(-\ heta_2)) \right]\n$$", "Using complex multiplication rules:\n- Real part: $ r_1 r_2 [\cos\ heta_1 \cos(-\ heta_2) - \sin\ heta_1 \sin(-\ heta_2)] = r_1 r_2 \cos(\ heta_1 + \ heta_2) $\n- Imaginary part: $ r_1 r_2 [\sin\ heta_1 (-\sin\ heta_2) + \cos\ heta_1 \cos(-\ heta_2)] = r_1 r_2 \sin(\ heta_1 - \ heta_2) $", "Thus,\n$$\nz_1 \overline{z_2} = r_1 r_2 \left[ \cos(\ heta_1 + \ heta_2) + i \sin(\ heta_1 - \ heta_2) \right]\n$$", "From this, the argument of $ z_1 \overline{z_2} $ is $ \arg(z_1 \overline{z_2}) = \ heta_1 - \ heta_2 $, modulo $ 360^\circ $, to maintain the principal value in $ [0^\circ, 360^\circ) $.", "However, in many geometric and physical applications—such as in navigation, wave interference, or rotational mechanics—the smallest positive argument, representing the minimal angular separation, is of primary interest. If $ \ heta_1 - \ heta_2 $ lies in $ (-180^\circ, 180^\circ) $, the difference $ \ heta_1 - \ heta_2 $ already gives the smallest angle between the two directions on the complex plane.", "Here, it is given that the smallest positive argument of $ z_1 \overline{z_2} $ is $ 60^\circ $, confirming that $ \ heta_1 - \ heta_2 = 60^\circ $ modulo $ 360^\circ $ represents the minimal angular displacement from $ z_2 $ toward $ z_1 $, consistent with standard angular conventions.", "Because $ |z_1 \overline{z_2}| = |z_1||z_2| $, and magnitude affects only position along the ray (not angle), the angle determines orientation uniquely in the $ [0^\circ, 180^\circ] $ range for the basic argument. Thus, the smallest positive argument—the principal value—lies in this interval and is mathematically defined as $ \ heta_1 - \ heta_2 $, reducible modulo $ 360^\circ $ to lie within $ [0^\circ, 360^\circ) $.", "In practice, recognizing that $ \arg(z_1 \overline{z_2}) = \ heta_1 - \ heta_2 \mod 360^\circ $ allows engineers and scientists to compute angular differences efficiently, especially when analyzing rotations, phase angles, or relative orientation—making this property both fundamental and widely applicable.", "Summary:\n- $ \arg(z_1 \overline{z_2}) = \ heta_1 - \ heta_2 \mod 360^\circ $\n- The smallest positive argument corresponds to the minimal angular separation, which is $ 60^\circ $ here\n- The cosine of this angle (or the cosine of the difference) determines uniqueness within $ [0^\circ, 180^\circ] $ for principal direction consistency\n- This principle supports practical computation in fields from signal processing to robotics", "Understanding this core identity helps clarify complex number multiplication's geometric meaning and supports accurate angular reasoning in both theory and application."]