If $a=2u+3v+7w$, $b=u+v-2w$ and $c=-u-2v-3w$, then $\left|\frac{[u, v, w]}{[a, b, c]}\right|(a+b+c) = $

If $2 \hat{i}-\hat{j}+3 \hat{k}$, $-12 \hat{i}-\hat{j}-3 \hat{k}$, $-\hat{i}+2 \hat{j}-4 \hat{k}$, and $\lambda \hat{i}+2 \hat{j}-\hat{k}$ are the position vectors of four coplanar points, then $\lambda=$

If $\vec{w} = \alpha (\vec{a} \times \vec{b}) + \beta (\vec{b} \times \vec{c}) + \gamma (\vec{c} \times \vec{a})$,$[\vec{a}, \vec{b}, \vec{c}] = 2$ and $\vec{w} \cdot (\vec{a} + \vec{b} + \vec{c}) = 8$,then $\alpha + \beta + \gamma =$

If $a, b,$ and $c$ are unit coplanar vectors,then the scalar triple product $[a, b, c]$ is equal to:

If the points having the position vectors $3 \hat{i}-2 \hat{j}-\hat{k}, 2 \hat{i}+3 \hat{j}-4 \hat{k}, -\hat{i}+\hat{j}+2 \hat{k}$ and $4 \hat{i}+5 \hat{j}+\lambda \hat{k}$ are coplanar,then $\lambda=$

If $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors and $\bar{p}=\frac{\bar{b} \times \bar{c}}{[\bar{a} \bar{b} \bar{c}]}, \bar{q}=\frac{\bar{c} \times \bar{a}}{[\bar{a} \bar{b} \bar{c}]}, \bar{r}=\frac{\bar{a} \times \bar{b}}{[\bar{a} \bar{b} \bar{c}]}$,then $\bar{a} \cdot \bar{p}+\bar{b} \cdot \bar{q}+\bar{c} \cdot \bar{r}=$