Concept:Equating the definitions of the scalar (dot) product and vector (cross) product magnitudes.
Formula:$$|\vec{A} \cdot \vec{B}| = AB \cos\theta$$
$$|\vec{A} \times \vec{B}| = AB \sin\theta$$
Solution:Set the two magnitudes equal to each other:
$$AB \cos\theta = AB \sin\theta$$
Since the vectors are non-zero (\(A \neq 0, B \neq 0\)), we can divide both sides by \(AB \cos\theta\):
$$\frac{\sin\theta}{\cos\theta} = 1 \implies \tan\theta = 1$$
$$\theta = \arctan(1) = 45^\circ$$
Why other options are incorrect:- \(30^\circ\): \(\sin(30^\circ) = 0.5\) while \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\) (they are not equal).
- \(60^\circ\): \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\) while \(\cos(60^\circ) = 0.5\) (they are not equal).
- \(90^\circ\): The dot product is zero, and the cross product is at its maximum magnitude.
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