If $\bar{a}=3 \hat{i}+\hat{j}-\hat{k}, \bar{b}=2 \hat{i}-\hat{j}+23 \hat{k}$ and $\bar{c}=7 \hat{i}-\hat{j}+23 \hat{k}$,then which of the following is valid?

If $a, b, c$ are any three coplanar unit vectors,then

Let the vectors $\overrightarrow{u}_1 = \hat{i} + \hat{j} + a\hat{k}$,$\overrightarrow{u}_2 = \hat{i} + b\hat{j} + \hat{k}$ and $\overrightarrow{u}_3 = c\hat{i} + \hat{j} + \hat{k}$ be coplanar. If the vectors $\overrightarrow{v}_1 = (a+b)\hat{i} + c\hat{j} + c\hat{k}$,$\overrightarrow{v}_2 = a\hat{i} + (b+c)\hat{j} + a\hat{k}$ and $\overrightarrow{v}_3 = b\hat{i} + b\hat{j} + (c+a)\hat{k}$ are also coplanar,then $6(a+b+c)$ is equal to $..............$.

$(a + b) \cdot (b + c) \times (a + b + c) = $

If $\bar{a}=\hat{i}+5 \hat{k}, \bar{b}=2 \hat{i}+3 \hat{k}, \bar{c}=4 \hat{i}-\hat{j}+2 \hat{k}$ and $\bar{d}=\hat{i}-\hat{j}$,then $(\bar{c}-\bar{a}) \cdot(\bar{b} \times \bar{d})=$

For any three non-zero vectors $\vec{r}_{1}, \vec{r}_{2}$ and $\vec{r}_{3}$,the determinant $\left| \begin{matrix} \vec{r}_{1} \cdot \vec{r}_{1} & \vec{r}_{1} \cdot \vec{r}_{2} & \vec{r}_{1} \cdot \vec{r}_{3} \\ \vec{r}_{2} \cdot \vec{r}_{1} & \vec{r}_{2} \cdot \vec{r}_{2} & \vec{r}_{2} \cdot \vec{r}_{3} \\ \vec{r}_{3} \cdot \vec{r}_{1} & \vec{r}_{3} \cdot \vec{r}_{2} & \vec{r}_{3} \cdot \vec{r}_{3} \end{matrix} \right| = 0$. Which of the following is false?