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 ($\theta$ is angle between u and v)

• 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 A Exactly two values of $\theta$

Given

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

$\theta$ is acute angle between $\hat{u}$ and $\hat{v}$,

$\left|\hat{u}\right|=1,\left|\hat{v}\right|=1$

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

$\Rightarrow 6\left|\sin \theta \right|=1\Rightarrow \sin \theta =\frac{1}{6}$

Hence, there are exactly two values of $\theta$ in $[0,\pi ]$ for which $2\hat{u}\times 3\hat{v}$ is unit vector.