Cross Product of Two Vectors

If $\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c}=\overrightarrow{0}$ and $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}=\lambda (\overrightarrow{b}\times \overrightarrow{c})$, then what is the value of $\lambda$?

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 $\overrightarrow{OB}=2\hat{i}+\hat{j}-3\hat{k}$ ?

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

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

$G$ is the centroid of triangle $ABC$ and $A_1$ and $B_1$ are the midpoints of sides $AB$ and $AC,$ respectively. If ${∆}_1$, be the area of quadrilateral $G{A}_{1}A{B}_1$ and $\Delta$ be the area of triangle $ABC,$ then $\frac{\Delta }{{\Delta }_1}$ is equal to:

Two adjacent sides of a parallelogram $ABCD$ are $2\hat{i}+4\hat{j}-5\hat{k}$ and $\hat{i}+2\hat{j}+3\hat{k}.$ Then the value of $\left|\overrightarrow{AC}\times \overrightarrow{BD}\right|$ is

$P\left(\overrightarrow{\mathit{p}}\right)$ and $Q\left(\overrightarrow{\mathit{q}}\right)$ are the position vectors of two fixed points and $R\left(\overrightarrow{r}\right)$ is the position vector of a variable point. If $R$ moves such that $\left(\overrightarrow{r}-\overrightarrow{p}\right)\times \left(\overrightarrow{r}-\overrightarrow{q}\right)=\overrightarrow{0},$ then the locus of $R$ is

Vector $\overrightarrow{c}$ is perpendicular to vectors $\overrightarrow{a}=(2,-3,1)$ and $\overrightarrow{b}=\left(1,-2,3\right)$ and satisfies the condition $\overrightarrow{c}·\left(\hat{i}+2\hat{j}-7\hat{k}\right)$ $=10.$ Then vector $\overrightarrow{c}$ is equal to

Let the unit vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ be perpendicular to each other and the unit vector $\overrightarrow{c}$ be inclined at an angle $\theta$ to both $\overrightarrow{a}$ and $\overrightarrow{b}$. If $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}+z\left(\overrightarrow{a}\times \overrightarrow{b}\right)$, then

The moment about the point $\hat{i}+2\hat{j}+3\hat{k}$ of a force represented by $\hat{i}+\hat{j}+\hat{k}$ acting through the point $2\hat{i}+3\hat{j}+\hat{k}$ is 

If $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{b}=\hat{i}+3j+5\hat{k}$ and $\overrightarrow{c}=7\hat{i}+9\hat{j}+11\hat{k}$, then the area of parallelogram having diagonals $\overrightarrow{a}+\overrightarrow{b}$ and $\overrightarrow{b}+\overrightarrow{c}$ is

Three non-coplanar vectors $\overrightarrow{a},\overrightarrow{b}$ and $\overrightarrow{c}$ are drawn from a common initial point. The angle between the plane passing through the terminal points of these vectors and the vector $\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{b}\times \overrightarrow{c}+\overrightarrow{c}\times \overrightarrow{a}$ is

If $\left|\overrightarrow{\mathrm{a}}\right|=4,\left|\overrightarrow{\mathrm{b}}\right|=2$ and the angle between $\overrightarrow{\mathrm{a}}$ and $\overrightarrow{\mathrm{b}}$ is $\frac{\mathrm{\pi }}{6}$ then ${\left|\overrightarrow{\mathrm{a}}\times \overrightarrow{\mathrm{b}}\right|}^2$ is equal to

For two particular vectors $\overrightarrow{A}$ and $\overrightarrow{B}$, it is known that  $\overrightarrow{A}\times \overrightarrow{B}=\overrightarrow{B}\times \overrightarrow{A}$. What must be true about the two vectors?