If the four points with position vectors $-2\hat{i}+\hat{j}+\hat{k}$, $\hat{i}+\hat{j}+\hat{k}$, $\hat{j}-\hat{k}$, and $\lambda\hat{j}+\hat{k}$ are coplanar, then $\lambda$ is equal to

Let $\vec{c}$ be a vector coplanar with the unit vectors $\vec{a}$ and $\vec{b}$, and let $\vec{d}$ be the unit vector perpendicular to $\vec{a}$, $\vec{b}$, and $\vec{c}$. If $[\vec{a} \vec{b} \vec{d}] \vec{c} - [\vec{a} \vec{b} \vec{c}] \vec{d} = \hat{i} + 2\hat{j} + 2\hat{k}$ and the angle between $\vec{a}$ and $\vec{b}$ is $30^{\circ}$, then $|\vec{c}| =$

If the vectors $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar,then $\frac{[\bar{a}+2\bar{b} \quad \bar{b}+2\bar{c} \quad \bar{c}+2\bar{a}]}{[\bar{a} \quad \bar{b} \quad \bar{c}]}=$

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,where $(a, b, c \neq 1)$,then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

If $(1,5,35), (7,5,5), (1, \lambda, 7)$ and $(2 \lambda, 1, 2)$ are coplanar,then the sum of all possible values of $\lambda$ is

If the vectors $\hat{\imath}+\hat{\jmath}+\hat{k}$,$\hat{\imath}-\hat{\jmath}+\hat{k}$ and $2\hat{\imath}+3\hat{\jmath}+m\hat{k}$ are coplanar,then $m=$