Let $\vec{c}$ be a vector coplanar with the unit vectors $\vec{a}$ and $\vec{b}$, and let $\vec{d}$ be the unit vector perpendicular to $\vec{a}$, $\vec{b}$, and $\vec{c}$. If $[\vec{a} \vec{b} \vec{d}] \vec{c} - [\vec{a} \vec{b} \vec{c}] \vec{d} = \hat{i} + 2\hat{j} + 2\hat{k}$ and the angle between $\vec{a}$ and $\vec{b}$ is $30^{\circ}$, then $|\vec{c}| =$

If $\bar{a}=3 \hat{\imath}+\hat{\jmath}-\hat{k}, \bar{b}=2 \hat{\imath}-\hat{\jmath}+7 \hat{k}$ and $\bar{c}=7 \hat{\imath}-\hat{\jmath}+23 \hat{k}$ are three vectors,then which of the following statements is true?

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 $\lambda=$

The altitude of the parallelepiped,whose coterminus edges are the vectors $\bar{a}=\hat{i}+\hat{j}+\hat{k}$,$\bar{b}=2\hat{i}+4\hat{j}-\hat{k}$,and $\bar{c}=\hat{i}+\hat{j}+3\hat{k}$,where $\bar{a}$ and $\bar{b}$ are the sides of the base of the parallelepiped,is:

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?

If the vectors $\hat{\imath}+\hat{\jmath}+\hat{k}$,$\hat{\imath}-\hat{\jmath}+\hat{k}$ and $2\hat{\imath}+3\hat{\jmath}+m\hat{k}$ are coplanar,then $m=$