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

If the vectors $i + 3j$,$5k$,and $\Pi - j$ are coplanar,find the value of $P$.

If $\overline{p}=\hat{i}+\hat{j}+\hat{k}$ and $\overline{q}=\hat{i}-2 \hat{j}+\hat{k}$. Then a vector of magnitude $5 \sqrt{3}$ units perpendicular to the vector $\overline{q}$ and coplanar with $\overline{p}$ and $\overline{q}$ is

If $[\bar{a} \bar{b} \bar{c}] = 4$,then the volume (in cubic units) of the parallelepiped with $\bar{a} + 2 \bar{b}, \bar{b} + 2 \bar{c}$ and $\bar{c} + 2 \bar{a}$ as coterminal edges,is

If $a, b, c$ are any three vectors and their reciprocal vectors are $a^{-1}, b^{-1}, c^{-1}$ such that $[a, b, c] \neq 0$,then $[a^{-1}, b^{-1}, c^{-1}]$ is equal to:

If the points $A(1,1,2), B(2,1, p), C(1,0,3)$ and $D(2,2,0)$ are coplanar,then the value of $p$ is