Question

If \(|\bar u|=3,|\bar v|=2\ and\ |\bar u\times \bar v|=3\), then the angle between \(\bar u\ and\ \bar v\) is equal to

• A. \(\frac{\pi}{4}\ or\ \frac{3\pi}{4}\)
• B. \(\frac{\pi}{6}\ or\ \frac{5\pi}{6}\)
• C. \(\frac{\pi}{3}\ or\ \frac{2\pi}{3}\)
• D. \(\frac{\pi}{2}\)
• E. 0

Answer 1

The Correct Option is B

The magnitude of the cross product of two vectors is given by: \[ |\vec{u} \times \vec{v}| = |\vec{u}| |\vec{v}| \sin \theta \] where $\theta$ is the angle between the two vectors. Substituting the given values: \[ 3 = 3 \times 2 \times \sin \theta \] \[ 3 = 6 \sin \theta \] \[ \sin \theta = \frac{1}{2} \] The possible values of $\theta$ for $\sin \theta = \frac{1}{2}$ are: \[ \theta = \frac{\pi}{6} \quad \text{or} \quad \theta = \frac{5\pi}{6} \]

So, the correct option is (B) : \(\frac{\pi}{6}\ or\ \frac{5\pi}{6}\)

Answer 2

We are given \(|\bar{u}| = 3\), \(|\bar{v}| = 2\), and \(|\bar{u} \times \bar{v}| = 3\).

We know that \(|\bar{u} \times \bar{v}| = |\bar{u}||\bar{v}|\sin\theta\), where \(\theta\) is the angle between \(\bar{u}\) and \(\bar{v}\).

Substituting the given values, we have:

\(3 = (3)(2)\sin\theta\)

\(3 = 6\sin\theta\)

\(\sin\theta = \frac{3}{6} = \frac{1}{2}\)

The values of \(\theta\) for which \(\sin\theta = \frac{1}{2}\) in the interval \([0, 2\pi)\) are \(\theta = \frac{\pi}{6}\) and \(\theta = \frac{5\pi}{6}\).

Therefore, the angle between \(\bar{u}\) and \(\bar{v}\) is \(\frac{\pi}{6}\) or \(\frac{5\pi}{6}\).