If $3 \hat{i}+3 \hat{j}+\sqrt{3} \hat{k}$,$\hat{i}+\hat{k}$,and $\sqrt{3} \hat{i}+\sqrt{3} \hat{j}+\lambda \hat{k}$ are coplanar,then $\lambda$ is equal to

Let $a, b, c$ be distinct non-negative numbers. If the vectors $a\hat{i} + a\hat{j} + c\hat{k}$,$\hat{i} + \hat{k}$,and $c\hat{i} + c\hat{j} + b\hat{k}$ lie in a plane,then $c$ is

The sum of the distinct real values of $\mu$,for which the vectors $\mu \hat{i} + \hat{j} + \hat{k}$,$\hat{i} + \mu \hat{j} + \hat{k}$,and $\hat{i} + \hat{j} + \mu \hat{k}$ are coplanar,is

If $\vec{a}, \vec{b}, \vec{c}$ are any three non-zero non-coplanar vectors and vectors $\vec{p} = \frac{\vec{b} \times \vec{c}}{[\vec{a} \vec{b} \vec{c}]}, \vec{q} = \frac{\vec{c} \times \vec{a}}{[\vec{a} \vec{b} \vec{c}]}, \vec{r} = \frac{\vec{a} \times \vec{b}}{[\vec{a} \vec{b} \vec{c}]}$,then $[\vec{p} \vec{q} \vec{r}] = ...$

For what value of $a$ is the volume of the parallelepiped formed by the vectors $\hat{i} + a\hat{j} + \hat{k}$,$\hat{j} + a\hat{k}$,and $a\hat{i} + \hat{k}$ minimum?

Let $\vec{a}$ be a vector which is perpendicular to the vector $3 \hat{i} + \frac{1}{2} \hat{j} + 2 \hat{k}$. If $\vec{a} \times (2 \hat{i} + \hat{k}) = 2 \hat{i} - 13 \hat{j} - 4 \hat{k}$,then the projection of the vector $\vec{a}$ on the vector $2 \hat{i} + 2 \hat{j} + \hat{k}$ is