Question

If $\hat{u}$ and $\hat{v}$ are unit vectors and $\theta$ is the acute angle between them, then $2\hat{u}\times 3\hat{v}$ is a unit vector for

• A. Exactly two values of $\theta$
• B. More than two values of $\theta$
• C. No value of $\theta$
• D. Exactly one value of $\theta$

Answer 1

The correct option is D Exactly one value of $\theta$

Let $(2\hat{u}\times 3\hat{v})$ is a unit vector.

$|2\hat{u}\times 3\hat{v}|=1$

$\Rightarrow 6\left|\hat{u}\right|\left|\hat{v}\right|\left|\sin \theta \right|=1$

$\Rightarrow \sin \theta =\frac{1}{6}$

Hence, there is exactly one value of $\theta$ for which $(2\hat{u}\times 3\hat{v})$ is a unit vector.