If $\overrightarrow{a} = \alpha \hat{i} + \beta \hat{j} + 3 \hat{k}$,$\overrightarrow{b} = -\beta \hat{i} - \alpha \hat{j} - \hat{k}$,and $\overrightarrow{c} = \hat{i} - 2 \hat{j} - \hat{k}$ such that $\overrightarrow{a} \cdot \overrightarrow{b} = 1$ and $\overrightarrow{b} \cdot \overrightarrow{c} = -3$,then $\frac{1}{3}((\vec{a} \times \vec{b}) \cdot \vec{c})$ is equal to ............

If $a, b$ and $c$ are non-coplanar vectors and the four points with position vectors $2a+3b-c$,$a-2b+3c$,$3a+4b-2c$ and $ka-6b+6c$ are coplanar,then $k=$

If $\bar{V} = 2\bar{i} + \bar{j} - \bar{k}$ and $\bar{W} = \bar{i} + 3\bar{k}$,and if $\bar{U}$ is a unit vector,then the maximum value of $[\bar{U} \bar{V} \bar{W}]$ is ...

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 $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number,then for what value of $\lambda$ does the equation $[\lambda(\vec{a} + \vec{b}), \lambda^2\vec{b}, \lambda\vec{c}] = [\vec{a}, \vec{b} + \vec{c}, \vec{b}]$ hold?

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?