# Let the vectors $$\overrightarrow a$$ and $$\overrightarrow b$$ be such that $$| \overrightarrow a | = 3$$ and $$|\overrightarrow b | = \frac{\large \sqrt 2}{\large 3}$$ , then $$\overrightarrow a \times \overrightarrow b$$ is a unit vector, if the angle between $$\overrightarrow a$$ and $$\overrightarrow b$$ is

$\begin{array} (A) \frac{\pi}{6} \quad & (B) \frac{\pi}{4} \quad & (C) \frac{\pi}{3} \quad &(D) \frac{\pi}{2} \end{array}$

Toolbox:
• $\overrightarrow a\times \overrightarrow b=|\overrightarrow a||\overrightarrow b|\sin\theta\; \hat n$ where $\hat n$ is the unit vector.
Step 1:
Given $|\overrightarrow a |=3$ and $|\overrightarrow b|=\large\frac{\sqrt 2}{3}$ and $\overrightarrow a\times\overrightarrow b$ is a unit vector.
$\overrightarrow a\times \overrightarrow b=|\overrightarrow a||\overrightarrow b|\sin\theta\; \hat n$----(1) where $\hat n$ is the unit vector.
But $\overrightarrow a\times \overrightarrow b$ is also a unit vector.
We know $|\overrightarrow a|=3$ and $|\overrightarrow b|=\large\frac{\sqrt 2}{3}$
Step 2:
Substitute these values in eq(1) we get