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 $(a \neq 1, b \neq 1, c \neq 1)$,then the value of $abc-(a+b+c)$ is:

If the points with position vectors $3i - 2j - k$,$2i + 3j - 4k$,$-i + j + 2k$,and $4i + 5j + \lambda k$ are coplanar,then $\lambda = \dots$

The volume of the tetrahedron whose vertices are $A(-1, 2, 3)$, $B(3, -2, 1)$, $C(p, 1, 3)$, and $D(-1, -2, 4)$ is $\frac{16}{3}$ cubic units. Find the value of $p$.

If $u, v$ and $w$ are three non-coplanar vectors,then $(u + v - w) \cdot [(u - v) \times (v - w)]$ equals

If $[\bar{a} \times \bar{b} \quad \bar{b} \times \bar{c} \quad \bar{c} \times \bar{a}] = \lambda [\bar{a} \quad \bar{b} \quad \bar{c}]^2$,then $\lambda$ is equal to

If $\bar{a} = \bar{i} - \bar{j}$,$\bar{b} = \bar{j} - \bar{k}$,$\bar{c} = \bar{k} - \bar{i}$ and $\bar{d}$ is a unit vector such that $\bar{a} \cdot \bar{d} = 0$ and $[\bar{b} \bar{c} \bar{d}] = 0$,then the vector $\bar{d} = ....$