If $\vec{\alpha}$ is a unit vector, $\vec{\beta}=\hat{i}+\hat{j}-\hat{k}$, and $\vec{\gamma}=\hat{i}+\hat{k}$, then the maximum value of $[\vec{\alpha} \vec{\beta} \vec{\gamma}]$ is

Let $\vec{OD} = \hat{i} + 2\hat{j} + 6\hat{k}$ and $\vec{CB} = -3\hat{i} - 2\hat{k}$ be the diagonals of the parallelogram $OBDC$. If $\vec{OA} = \hat{i} + 2\hat{j} + 3\hat{k}$, then the volume of the parallelepiped determined by vectors $\vec{OA}, \vec{OB}$, and $\vec{OC}$ (in cubic units) is:

If the vectors $2\hat{i} - \hat{j} + \hat{k}$,$\hat{i} + 2\hat{j} - 3\hat{k}$,and $3\hat{i} + a\hat{j} + 5\hat{k}$ are coplanar,then the value of $a$ is:

Let $a = i - k$, $b = xi + j + (1 - x)k$, and $c = yi + xj + (1 + x - y)k$. Then $[a\,b\,c]$ depends on

If the volume of a tetrahedron having $\bar{i}+2 \bar{j}-3 \bar{k}$, $2 \bar{i}+\bar{j}-3 \bar{k}$, and $3 \bar{i}-\bar{j}+p \bar{k}$ as its coterminous edges is $2$, then the values of $p$ are the roots of the equation

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number,then for what value of $\lambda$ are the vectors $\vec{a} + 2\vec{b} + 3\vec{c}$,$\lambda\vec{b} + 4\vec{c}$,and $(2\lambda - 1)\vec{c}$ non-coplanar?