Question

The angle between two vectors \( \vec{a} \) and \( \vec{b} \) is 0 and \( |\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}| \) is given. Find the value of \( \theta \).

Hint:

When dot and cross product magnitudes are equal, the angle between the vectors is \( 45^\circ \).

Answer 1

The dot product is given by: \[ \vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta. \] The cross product magnitude is: \[ |\vec{a} \times \vec{b}| = |\vec{a}||\vec{b}|\sin\theta. \] Given that \( |\vec{a} \cdot \vec{b}| = |\vec{a} \times \vec{b}| \), we equate: \[ |\vec{a}||\vec{b}|\cos\theta = |\vec{a}||\vec{b}|\sin\theta. \] Dividing both sides by \( |\vec{a}||\vec{b}| \) (assuming nonzero vectors), \[ \cos\theta = \sin\theta. \] Solving \( \tan\theta = 1 \), we get: \[ \theta = 45^\circ \text{ or } \frac{\pi}{4}. \]