Cross Product of Two Vectors

If $\overrightarrow{\mathit{a}}$ and $\overrightarrow{b}$ are vectors such that $\left|\overrightarrow{a}\right|=2,$ $\left|\overrightarrow{b}\right|=7$ and $\overrightarrow{a}\times \overrightarrow{b}=3\hat{i}+2\hat{j}+6\hat{k},$ then what is the acute angle between $a$ and $b$?

Which one of the following is the unit vector perpendicular to both $\overrightarrow{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$?

What is the area of $\Delta OAB$, where $O$ is the origin such that $\overrightarrow{OA}=3\hat{i}-\hat{j}+\hat{k}$ and $\overrightarrow{OB}=2\hat{i}+\hat{j}-3\hat{k}$ ?

What is the vector perpendicular to both the vectors $\hat{i}-\hat{j}$ and $\hat{i}$?

If the magnitude of $\overrightarrow{a}\times \overrightarrow{b}$ equals to $\overrightarrow{a}·\overrightarrow{b}$, then which one of the following is correct?

Consider the following statements:

$\text{I}$. $4\hat{i}\times 3\hat{i}=0$

$\text{II}$. $\frac{4\hat{i}}{3\hat{i}}=\frac{4}{3}$

Which one of the above statement(s) is / are correct?

What is the value of $\lambda$ for which $\left(\lambda \hat{i}+\hat{j}-\hat{k}\right)\times \left(3\hat{i}-2\hat{j}+4\hat{k}\right)=\left(2\hat{i}-11\hat{j}-7\hat{k}\right)$.

If $\left|\overrightarrow{a}\right|=10, \left|\overrightarrow{b}\right|=2$ and $\overrightarrow{a}·\overrightarrow{b}=12$, then what is the value of $\left|\overrightarrow{a}\times \overrightarrow{b}\right|$?

If $\overrightarrow{a}=2\hat{i}-3\hat{j}-\hat{k}, \overrightarrow{b}=\hat{i}+4\hat{j}-2\hat{k}$, then what is $(\overrightarrow{a}+\overrightarrow{b})\times (\overrightarrow{a}-\overrightarrow{b})$ is equal to?

If $\overrightarrow{u}=\overrightarrow{a}-\overrightarrow{b}, \overrightarrow{v}=\overrightarrow{a}+\overrightarrow{b}$ and $\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=2$, then  $\left|\overrightarrow{u}\times \overrightarrow{v}\right|$ is equal to

If $\overrightarrow{a}$ be any vector, then ${\left|\overrightarrow{a}\times \hat{i}\right|}^2+{\left|\overrightarrow{a}\times \hat{j}\right|}^2+{\left|\overrightarrow{a}\times \hat{k}\right|}^2$ is equal to

The area of parallelogram whose adjacent sides are $\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k}, \overrightarrow{b}=3\hat{i}-2\hat{j}+\hat{k}$ is

If $\left(3\overrightarrow{a}-\overrightarrow{b}\right)\times \left(\overrightarrow{a}+3\overrightarrow{b}\right)=k\overrightarrow{a}\times \overrightarrow{b}$, then what is the value of $k$$?$