If $\bar{a}, \bar{b}, \bar{c}$ are mutually perpendicular vectors such that $|\bar{a}| = a, |\bar{b}| = b, |\bar{c}| = c$,then $[\bar{a} \bar{b} \bar{c}] = ......$

If the four points,whose position vectors are $3 \hat{i} - 4 \hat{j} + 2 \hat{k}$,$\hat{i} + 2 \hat{j} - \hat{k}$,$-2 \hat{i} - \hat{j} + 3 \hat{k}$,and $5 \hat{i} - 2 \alpha \hat{j} + 4 \hat{k}$ are coplanar,then $\alpha$ is equal to

If the volume of the parallelepiped formed by three non-coplanar vectors $\vec{a}, \vec{b}$ and $\vec{c}$ is $4$ cubic units,then $[\vec{a} \times \vec{b} \quad \vec{b} \times \vec{c} \quad \vec{c} \times \vec{a}]$ is equal to

Let $v = 2i + j - k$ and $w = i + 3k$. If $u$ is any unit vector, then the maximum value of the scalar triple product $[u v w]$ is

If the vectors $2i + j - k$,$-i + 2j + \lambda k$,and $-5i + 2j - k$ are coplanar,then the value of $\lambda$ is equal to:

If $(2,3,9), (5,2,1), (1, \lambda, 8)$ and $(\lambda, 2,3)$ are coplanar,then the product of all possible values of $\lambda$ is.