Scalar TripIe Product

$\left(\overline{a}\times \overline{b}\right)\times \left(\overline{a}\times \overline{c}\right)=$

Let the vectors $\overrightarrow{u_1}=\hat{i}+\hat{j}+a\hat{k}, \overrightarrow{u_2}=\hat{i}+b\hat{j}+\hat{k},$ and $\overrightarrow{u_3}=c\hat{i}+\hat{j}+\hat{k}$ be coplanar. If the vectors $\overrightarrow{v_1}=\left(a+b\right)\hat{i}+c\hat{j}+c\hat{k}, \overrightarrow{v_2}=a\hat{i}+\left(b+c\right)\hat{j}+a\hat{k}$ and  ${\overrightarrow{v}}_3=b\hat{i}+b\hat{j}+\left(c+a\right)\hat{k}$ are also coplanar, then $6\left(a+b+c\right)$ is equal to 

Let $\overrightarrow{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\overrightarrow{b}=\hat{i}+\hat{j}-\hat{k}$. If $\overrightarrow{c}$ is a vector such that $\overrightarrow{a}·\overrightarrow{c}=11,\overrightarrow{b}·\left(\overrightarrow{a}\times \overrightarrow{c}\right)=27$ and $\overrightarrow{b}·\overrightarrow{c}=$ $-\sqrt{3}\left|\overrightarrow{b}\right|$, then $|\overrightarrow{a}\times \overrightarrow{c}{|}^2$ is equal to

If four distinct points with position vectors $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}$ and $\overrightarrow{d}$ are coplanar, then $\left[\overrightarrow{a}\overrightarrow{b}\overrightarrow{c}\right]$ is equal to

Let $a, b, c$ be three distinct real numbers, none equal to one. If the vectors $a\hat{i}+\hat{j}+\hat{k}, \hat{i}+b\hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+c\hat{k}$ are coplanar, then $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is equal to

The sum of all values of $\alpha$, for which the points whose position vectors are $\hat{\mathrm{i}}-2\hat{\mathrm{j}}+3\hat{\mathrm{k}}, 2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}}, \left(\alpha +1\right)\hat{\mathrm{i}}+2\hat{\mathrm{k}}$ and $9\hat{\mathrm{i}}+\left(\alpha -8\right)\hat{\mathrm{j}}+6\hat{\mathrm{k}}$ are coplanar, is equal to

Let the vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$ represent three coterminous edges of a parallelopiped of volume $V$. Then the volume of the parallelopiped, whose coterminous edges are represented by $\overrightarrow{a}, \overrightarrow{b}+\overrightarrow{c}$ and $\overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c}$ is equal to

Let the vectors $\overrightarrow{u_1}=\hat{i}+\hat{j}+a\hat{k}, \overrightarrow{u_2}=\hat{i}+b\hat{j}+\hat{k},$ and $\overrightarrow{u_3}=c\hat{i}+\hat{j}+\hat{k}$ be coplanar. If the vectors $\overrightarrow{v_1}=\left(a+b\right)\hat{i}+c\hat{j}+c\hat{k}, \overrightarrow{v_2}=a\hat{i}+\left(b+c\right)\hat{j}+a\hat{k}$ and  ${\overrightarrow{v}}_3=b\hat{i}+b\hat{j}+\left(c+a\right)\hat{k}$ are also coplanar, then $6\left(a+b+c\right)$ is equal to 

If $a\hat{i}+\hat{j}+\hat{k}, \hat{i}+b\hat{j}+\hat{k}, \hat{i}+\hat{j}+c\hat{k}$ are coplanar then the value $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

If $\overrightarrow{a},\overrightarrow{b},\overrightarrow{c},\overrightarrow{d}$ are coplanar vectors then the value of $\left[\begin{array}{ccc}\overrightarrow{a}& \overrightarrow{b}& \overrightarrow{c}\end{array}\right]$ is

If $V$ is volume of parallelepiped whose edges determined by vectors $\overrightarrow{a}, \overrightarrow{b}, \overrightarrow{c}$, then volume of parallelepiped whose edges determined by vectors $\overrightarrow{a}, \overrightarrow{a}+\overrightarrow{b}, \overrightarrow{a}+2\overrightarrow{b}+3\overrightarrow{c}$ is

Sum of all values of $\alpha$ for which $\hat{\mathrm{i}}-2\hat{\mathrm{j}}+3\hat{\mathrm{k}}, 2\hat{\mathrm{i}}-3\hat{\mathrm{j}}+4\hat{\mathrm{k}}, \left(\mathrm{\alpha }+1\right)\hat{\mathrm{i}}+2\hat{\mathrm{k}} \mathrm{and} 9\hat{\mathrm{i}}+\left(\mathrm{\alpha }-8\right)\hat{\mathrm{j}}+6\hat{\mathrm{k}}$ are coplanar.

Let $\overrightarrow{u}=\hat{i}-\hat{j}-2\hat{k}$, $\overrightarrow{v}=2\hat{i}+\hat{j}-\hat{k}$, $\overrightarrow{v}·\overrightarrow{w}=2$ and $\overrightarrow{v}\times \overrightarrow{w}=\overrightarrow{u}+\lambda \overrightarrow{v}$, then $\overrightarrow{u}·\overrightarrow{w}$ is equal to

Let $\overrightarrow{p}, \overrightarrow{q}, \overrightarrow{r}, \overrightarrow{s}$  be the vectors such that $\overrightarrow{s}=x\left(\overrightarrow{p}\times \overrightarrow{q}\right)+y\left(\overrightarrow{q}\times \overrightarrow{r}\right)+z\left(\overrightarrow{r}\times \overrightarrow{p}\right)$. If $\left[\begin{array}{ccc}\overrightarrow{p}& \overrightarrow{q}& \overrightarrow{r}\end{array}\right]=8$ and $s·\left[\overrightarrow{p}+\overrightarrow{q}+\overrightarrow{r}\right]=16$, then $x+y+z$ is equal to

Let $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}, \overrightarrow{b}=\hat{i}-2\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}+3\hat{j}-2\hat{k},\overrightarrow{d}=2\hat{i}+\hat{j}-\hat{k}$ be four vectors and let $l=\overrightarrow{b}·\overrightarrow{c}$ and $m=\overrightarrow{b}·\overrightarrow{a}$. Then $\left[\begin{array}{ccc}\left(m\overrightarrow{b}+I\overrightarrow{a}\right)& \overrightarrow{b}& \overrightarrow{d}\end{array}\right]$

Let $\overrightarrow{a}$ be a vector in the plane containing vectors $\overrightarrow{b}=\hat{i}+2\hat{j}+\hat{k}$ and $\overrightarrow{c}=2\hat{i}-\hat{j}+\hat{k}$. If $\overrightarrow{a}$ is perpendicular to $\hat{i}+\hat{j}+3\hat{k}$ and its projection on $\overrightarrow{b}$ is $3\sqrt{6}$, then ${\left|\overrightarrow{a}\right|}^2=$

What is the value of $\left(\overrightarrow{A}+\overrightarrow{B}\right)·\left(\overrightarrow{A}\times \overrightarrow{B}\right)$?

let a vector $\overrightarrow{c}$ be coplanar with the vectors $\overrightarrow{a}=-\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=2\hat{i}+\hat{j}-\hat{k}$. If the vector $\overrightarrow{c}$ also satisfies the conditions $\overrightarrow{c}·\left[\left(\overrightarrow{a}+\overrightarrow{b}\right)\times \left(\overrightarrow{a}\times \overrightarrow{b}\right)\right]=-42$ and $\left(\overrightarrow{c}\times \left(\overrightarrow{a}-\overrightarrow{b}\right)\right)·\hat{k}=3$, then the value of ${\left|\overrightarrow{c}\right|}^2$ is equal to

Consider a vector$\overrightarrow{r}=\overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{b}$ . where the projection of $\overrightarrow{r}$on $\overrightarrow{a}$ is $2$. Then the maximum value of ${\left|\overrightarrow{r}\right|}^2=$