If the volume of the parallelepiped formed by the vectors $\hat{i} + \lambda \hat{j} + \hat{k}$,$\hat{j} + \lambda \hat{k}$ and $\lambda \hat{i} + \hat{k}$ is minimum,then $\lambda$ is equal to

If the points $2a+3b-c, a-2b+3c, 3a+\lambda b-2c$ and $a-6b+6c$ are coplanar,then the direction cosines of the vector $\lambda \hat{i}-2\lambda \hat{j}+\hat{k}$ are

If the vectors $ai + j + k$,$i + bj + k$,and $i + j + ck$ $(a \ne 1, b \ne 1, c \ne 1)$ are coplanar,then the value of $\frac{1}{1 - a} + \frac{1}{1 - b} + \frac{1}{1 - c} = $

If $\vec{a}=\hat{i}-\hat{k}, \vec{b}=x \hat{i}+\hat{j}+(1-x) \hat{k}$ and $\vec{c}=y \hat{i}+x \hat{j}+(1+x-y) \hat{k}$,then $[\vec{a} \vec{b} \vec{c}]$ depends on

$a \cdot [(b + c) \times (a + b + c)]$ is equal to

If three non-zero vectors are $a = a_1 i + a_2 j + a_3 k,$ $b = b_1 i + b_2 j + b_3 k$ and $c = c_1 i + c_2 j + c_3 k.$ If $c$ is the unit vector perpendicular to the vectors $a$ and $b$ and the angle between $a$ and $b$ is $\frac{\pi}{6},$ then $\left| \begin{array}{ccc} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{array} \right|^2$ is equal to