If $\vec{a} = 2\hat{i} + \hat{j} - \hat{k}$, $\vec{b} = \hat{i} + 3\hat{k}$ and $\vec{c}$ is a unit vector, then the maximum value of the scalar triple product $[\vec{a} \vec{b} \vec{c}]$ is

Let $\vec{p} = 2\hat{i} + 3\hat{j} + a\hat{k}$,$\vec{q} = b\hat{i} + 5\hat{j} - \hat{k}$,and $\vec{r} = \hat{i} + \hat{j} + 3\hat{k}$. If $\vec{p}, \vec{q}, \vec{r}$ are coplanar and $\vec{p} \cdot \vec{q} = 20$,then the ordered pair $(a, b)$ is:

If vectors $\vec{A} = 2i + 3j + 4k$,$\vec{B} = i + j + 5k$,and $\vec{C}$ form a left-handed system,then $\vec{C}$ is

If $\overline{a}=2 \hat{\imath}-\hat{\jmath}+\hat{k}$,$\overline{b}=\hat{\imath}+2 \hat{\jmath}-3 \hat{k}$ and $\overline{c}=3 \hat{\imath}+\lambda \hat{\jmath}+5 \hat{k}$ are coplanar,then $\lambda$ is the root of the equation

Let $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + 3\hat{k}$. For what value of $\lambda$ is the vector $\vec{c} = \lambda\hat{i} + \hat{j} + (2\lambda - 1)\hat{k}$ parallel to the plane containing $\vec{a}$ and $\vec{b}$?

If the vertices of a tetrahedron are $\vec{a} = \vec{j} + 2\vec{k}$,$\vec{b} = 3\vec{i} + \vec{k}$,$\vec{c} = 4\vec{i} + 3\vec{j} + 6\vec{k}$,and $\vec{d} = 2\vec{i} + 3\vec{j} + 2\vec{k}$,find its volume.