If $p = \frac{b \times c}{[a, b, c]}, q = \frac{c \times a}{[a, b, c]}, r = \frac{a \times b}{[a, b, c]}$,where $a, b, c$ are three non-coplanar vectors,then the value of $(a + b + c) \cdot (p + q + r)$ is given by

Let $\overrightarrow{A} = \hat{i} + \hat{j} + \hat{k}$,$\overrightarrow{B} = \hat{i}$,and $\overrightarrow{C} = C_1\hat{i} + C_2\hat{j} + C_3\hat{k}$. If $C_2 = -1$ and $C_3 = 1$,then to make the three vectors coplanar:

If the volume of the tetrahedron whose coterminous edges are given by the vectors $\vec{a} = -2\hat{i} + 3\hat{j} - 3\hat{k}$, $\vec{b} = 4\hat{i} + 5\hat{j} + (\lambda - 10)\hat{k}$, and $\vec{c} = 6\hat{i} + 2\hat{j} - 3\hat{k}$ is $11$ cubic units, then the sum of the possible values of $\lambda$ is

If $A$,$B$,and $C$ are three non-coplanar vectors,then $(A + B + C) \cdot ((A + B) \times (A + C)) = \dots$

If the vectors $\vec{a}=\lambda \hat{i}+\mu \hat{j}+4 \hat{k}$,$\vec{b}=2 \hat{i}+4 \hat{j}-2 \hat{k}$ and $\vec{c}=2 \hat{i}+3 \hat{j}+\hat{k}$ are coplanar and the projection of $\vec{a}$ on the vector $\vec{b}$ is $\sqrt{54}$ units,then the sum of all possible values of $\lambda+\mu$ is equal to:

If three conterminous edges of a parallelepiped are represented by $\vec{a} - \vec{b}$,$\vec{b} - \vec{c}$,and $\vec{c} - \vec{a}$,then its volume is