Concept:Calculating the magnitude of a cross product using the angle between two vectors.
Formula:$$|\vec{R}_1 \times \vec{R}_2| = R_1 R_2 \sin\theta$$
Solution:First, determine the angle \(\theta\) between the two vectors:
$$\theta = \theta_2 - \theta_1 = 90^\circ - 30^\circ = 60^\circ$$
Now substitute the magnitudes and the angle into the formula:
$$|\vec{R}_1 \times \vec{R}_2| = (4\text{ cm})(3\text{ cm}) \sin(60^\circ)$$
$$|\vec{R}_1 \times \vec{R}_2| = 12 \times \frac{\sqrt{3}}{2} = 6\sqrt{3}\text{ cm}^2$$
Why other options are incorrect:- A is incorrect because it omits the factor of \(1/2\) from the sine value (using 1 instead of \(\sin 60^\circ\)).
- C and D are mathematically incorrect results from applying the wrong trigonometric functions or angles.
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