If $a, b, c$ are three non-coplanar vectors,then $\frac{a \cdot (b \times c)}{c \times a \cdot b} + \frac{b \cdot (a \times c)}{c \cdot (a \times b)} = $

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

The sum of the distinct real values of $\mu$,for which the vectors $\mu \hat{i} + \hat{j} + \hat{k}$,$\hat{i} + \mu \hat{j} + \hat{k}$,and $\hat{i} + \hat{j} + \mu \hat{k}$ are coplanar,is

If the origin $O(0,0,0)$ and the points $P(2,3,4)$,$Q(1,2,3)$,and $R(x, y, z)$ are co-planar,then:

If the vectors $a \hat{i}+\hat{j}+\hat{k}, \hat{i}+b \hat{j}+\hat{k}, \hat{i}+\hat{j}+c \hat{k}$ $(a \neq 1, b \neq 1, c \neq 1)$ are coplanar,then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}$ is

If $a, b$ and $c$ are three non-coplanar vectors,then $(a + b + c) \cdot [(a + b) \times (a + c)]$ is equal to