The volume of the tetrahedron having the edges $\hat{i}+2\hat{j}-\hat{k}$, $\hat{i}+\hat{j}+\hat{k}$, and $\hat{i}-\hat{j}+\lambda\hat{k}$ as coterminous edges is $\frac{2}{3}$ cubic units. Then $\lambda$ equals:

If for vectors $\bar{a}, \bar{b},$ and $\bar{c},$ $[\bar{a} \bar{b} \bar{c}] = 4,$ then $[\bar{a} \times \bar{b}, \bar{b} \times \bar{c}, \bar{c} \times \bar{a}] = \dots$

If $a=2 \hat{i}-3 \hat{j}+5 \hat{k}$, $b=3 \hat{i}-4 \hat{j}+5 \hat{k}$ and $c=5 \hat{i}-3 \hat{j}-2 \hat{k}$, then the volume of the parallelepiped with coterminous edges $a+b$, $b+c$, $c+a$ is

For what value of $a$ is the volume of the parallelepiped formed by the vectors $i + aj + k$,$j + ak$,and $ai + k$ minimum?

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 $\frac{[\vec{b} \times \vec{c}, \vec{c} \times \vec{d}, \vec{d} \times \vec{b}]}{[\vec{b}+\vec{c}, \vec{c}+\vec{d}, \vec{d}+\vec{b}]}$.

If $a, b, c$ are three non-coplanar vectors and $d$ is any unit vector, then $|(a \cdot d)(b \times c) + (b \cdot d)(c \times a) + (c \cdot d)(a \times b)| = $