Vector Algebra

If

 $\left[\overrightarrow{a}\overrightarrow{,b}\overrightarrow{,c}\right]=1\$  then the value of  $\ \frac{\overrightarrow{a}\cdot\left(\overrightarrow{b}\times\overrightarrow{c}\right)}{\left(\overrightarrow{c}\times\overrightarrow{a}\right)\cdot\overrightarrow{b}}\ +\frac{\overrightarrow{b}\cdot\left(\overrightarrow{c}\times\overrightarrow{a}\right)}{\left(\overrightarrow{a}\times\overrightarrow{b}\right)\cdot\overrightarrow{c}}+\frac{\overrightarrow{c}\cdot\left(\overrightarrow{a}\times\overrightarrow{b}\right)}{\left(\overrightarrow{c}\times\overrightarrow{b}\right)\cdot\overrightarrow{a}}$  

If  $\ \overrightarrow{a}\ and\ \overrightarrow{b}\$ are the unit vectors such that    $\left[\overrightarrow{a},\overrightarrow{b},\ \overrightarrow{a}\times\overrightarrow{b}\right]=\frac{\pi}{4},\$  then the angle between   $\ \overrightarrow{a}\ and\ \overrightarrow{b}\$  is 

 Consider the vectors $\ \overrightarrow{a},\overrightarrow{b},\overrightarrow{c},\overrightarrow{d}$ such that   $\ \left(\overrightarrow{a}\times\overrightarrow{b}\right)\times\left(\overrightarrow{c}\times\overrightarrow{d}\right)=\overrightarrow{0}.$  Let  $P_1$   and  $P_2$   be the planes determined  by the pairs of vectors $\overrightarrow{a},\overrightarrow{b}\ and\ \overrightarrow{c},\overrightarrow{d}\$  respectively .Then the angle between $P_1\ and\ P_2\ is\$   

If  $\ \overrightarrow{a}\times\left(\overrightarrow{b}\times\overrightarrow{c}\right)=\left(\overrightarrow{a}\times\overrightarrow{b}\right)\times\overrightarrow{c},$   $\ where\ \overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\$ are any three vectors such that  $\overrightarrow{b}\cdot\overrightarrow{c}\ne0\ and\ \overrightarrow{a}\cdot\overrightarrow{b}\ne0,\ then\ \overrightarrow{a}\ and\ \overrightarrow{b\ }\ are\$  

If  $\ \overrightarrow{a},\ \overrightarrow{b},\overrightarrow{c}\$  are non - coplanar ,non-zero vectors such that  $\ \left[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\right]=3\ then\ \left\{\left[\overrightarrow{a}\times\overrightarrow{b},\ \overrightarrow{b}\times\overrightarrow{c},\overrightarrow{c}\times\overrightarrow{a}\right]\right\}^2\$  is equal to 

(a)  

If the volume of the parallelpiped with   $\ \left(\overrightarrow{a}\times\overrightarrow{b}\right)\times\left(\overrightarrow{b}\times\overrightarrow{c}\right),\left(\overrightarrow{b}\times\overrightarrow{c}\right)\times\left(\overrightarrow{c}\times\overrightarrow{a}\right)\ and\$   $\ \left(\overrightarrow{c}\times\overrightarrow{a}\right)\times\left(\overrightarrow{a}\times\overrightarrow{b}\right)$  as conterminous edges is ,

(a)  

If the length of the perpendicular form from the origin to the plane $2x+3y+\lambda z=1,\ \lambda>0\ is\ \frac{1}{5},\$   $then\ the\ value\ of\ \lambda\ is\$  

The distance between the planes  $x+2y+3z+7=0\ \$  and  $2x+4y+6z+7=0$   is 

If the line $\frac{x-2}{3}=\frac{y-1}{-5}=\frac{z+2}{2}$  lines in the plane

 $x+3y-\alpha z+\beta=0,$  then  $\left(\alpha,\beta\right)$  is