Kerala KEAM 5 June 2024 Solved Paper

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