If $\overline{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\overline{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$,then the value of $(2 \bar{a}-\bar{b}) \cdot [(\bar{a} \times \bar{b}) \times (\bar{a}+2 \bar{b})] = $

If $35 \hat{i}+14 \hat{j}-77 \hat{k}$,$2 \hat{i}+7 \hat{j}+5 \hat{k}$ and $5 \hat{i}+2 \hat{j}+\lambda \hat{k}$ are coplanar,then $\lambda$ is equal to

If $\bar{x}=\frac{\bar{b} \times \bar{c}}{[\bar{a} \bar{b} \bar{c}]}, \bar{y}=\frac{\bar{c} \times \bar{a}}{[\bar{a} \bar{b} \bar{c}]}$ and $\bar{z}=\frac{\bar{a} \times \bar{b}}{[\bar{a} \bar{b} \bar{c}]}$ where $\bar{a}, \bar{b}, \bar{c}$ are non-coplanar vectors,then the value of $\bar{x} \cdot(\bar{a}+\bar{b})+\bar{y} \cdot(\bar{b}+\bar{c})+\bar{z} \cdot(\bar{c}+\bar{a})$ is

The points $A(4, 5, 1)$,$B(0, -1, -1)$,$C(3, 9, 4)$,and $D(-4, 4, 4)$ are:

Observe the following statements:
$A$. Three vectors are coplanar if one of them is expressible as a linear combination of the other two.
$R$. Any three coplanar vectors are linearly dependent.
Then, which of the following is true?

The vector $c \cdot (b+c) \times (a+b+c)$ is equal to