If $i, j, k$ are the unit vectors and mutually perpendicular,then $[i, k, j]$ is equal to

$[(\vec{a} \times \vec{b}) \times (\vec{a} \times \vec{c})] \cdot \vec{d} = \dots$

The volume (in cubic units) of the tetrahedron with edges $\hat{i}+\hat{j}+\hat{k}$, $\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}-\hat{k}$ is

The value of $m$,if the vectors $\hat{\imath}-\hat{\jmath}-6 \hat{k}$,$\hat{\imath}-3 \hat{\jmath}+4 \hat{k}$,and $2 \hat{\imath}-5 \hat{\jmath}+m \hat{k}$ are coplanar,is

Consider the four points $A(1, -2, -1)$, $B(4, 0, -3)$, $C(1, 2, -1)$, and $D(2, -4, -5)$ in space. If $\vec{b} = \vec{AB}$, $\vec{c} = \vec{AC}$, and $\vec{d} = \vec{AD}$, then find the value of $\frac{[\vec{b} \times \vec{c}, \vec{c} \times \vec{d}, \vec{d} \times \vec{b}]}{[\vec{b}+\vec{c}, \vec{c}+\vec{d}, \vec{d}+\vec{b}]}$.

$(a + b) \cdot (b + c) \times (a + b + c) = $