Question

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

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

Answer 1

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

Given,

$\overline{u}$ and $\overrightarrow{v}$ are unit vector

$|\overrightarrow{u}|=1$

$|\overrightarrow{v}|=1$

Now,

$\Rightarrow |2\overrightarrow{u}||\overrightarrow{3v}|\sin \theta =1$

$|\overrightarrow{2u}\times \to \overrightarrow{3v}|$

$[\because |\overrightarrow{a}\times \overrightarrow{b}|=|\overrightarrow{a}|\overrightarrow{b}||\sin \theta ]$

$\Rightarrow 2\cdot 3|\overline{u}|\nabla \mid \sin \theta =1$

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

$\text{There is only one value of}\theta$

$\text{for which}|\overrightarrow{2}4\times \overrightarrow{3}V|\text{is a}$

$\text{unit vector.}$

$\text{option}B$