Question

If \( \overrightarrow{a} \) and \( \overrightarrow{b} \) are two nonzero vectors and if \( |\overrightarrow{a} \times \overrightarrow{b}| = |\overrightarrow{a} \cdot \overrightarrow{b}| \), then the angle between \( \overrightarrow{a} \) and \( \overrightarrow{b} \) is equal to

• A. \( \frac{\pi}{2} \)
• B. \( \frac{\pi}{4} \)
• C. \( \frac{\pi}{3} \)
• D. 6
• E. \( \frac{2\pi}{3} \)

Hint:

When the magnitude of the cross product equals the magnitude of the dot product, the angle between the vectors is \( \frac{\pi}{4} \).

Answer 1

The Correct Option is B

We are given two nonzero vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \), and the condition: \[ |\overrightarrow{a} \times \overrightarrow{b}| = |\overrightarrow{a} \cdot \overrightarrow{b}|. \] The magnitude of the cross product of two vectors is given by: \[ |\overrightarrow{a} \times \overrightarrow{b}| = |\overrightarrow{a}| |\overrightarrow{b}| \sin \theta, \] where \( \theta \) is the angle between the vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \). The magnitude of the dot product of two vectors is given by: \[ |\overrightarrow{a} \cdot \overrightarrow{b}| = |\overrightarrow{a}| |\overrightarrow{b}| \cos \theta. \] Since the magnitudes of the vectors \( \overrightarrow{a} \) and \( \overrightarrow{b} \) are nonzero, we can divide both sides of the equation by \( |\overrightarrow{a}| |\overrightarrow{b}| \), giving: \[ \sin \theta = \cos \theta. \] This equation holds when: \[ \theta = \frac{\pi}{4}. \]
Thus, the angle between \( \overrightarrow{a} \) and \( \overrightarrow{b} \) is \( \frac{\pi}{4} \), and the correct answer is option (B).