Let $V = 2i + j - k$ and $W = i + 3k$. If $U$ is a unit vector,then the maximum value of the scalar triple product $[U V W]$ is

If three vectors $\vec{a} = 12\hat{i} + 4\hat{j} + 3\hat{k}$,$\vec{b} = 8\hat{i} - 12\hat{j} - 9\hat{k}$,and $\vec{c} = 33\hat{i} - 4\hat{j} - 24\hat{k}$ represent the coterminous edges of a parallelepiped,then its volume is:

Let $\vec{a} = 2\hat{i} - 3\hat{j} + \hat{k}$, $\vec{b} = 3\hat{i} + 2\hat{j} + 2\hat{k}$, and $\vec{c} = 4\hat{i} - 3\hat{j} + \hat{k}$. The vectors $\vec{a}, \vec{b}, \text{ and } \vec{c}$ are:

If the origin and the points $(1, 2, 3)$,$(2, 3, 4)$,and $(x, y, z)$ are coplanar,then

If $\hat{a}=\frac{1}{\sqrt{10}}(3 \hat{i}+\hat{k})$ and $\hat{b}=\frac{1}{7}(2 \hat{i}+3 \hat{j}-6 \hat{k})$,then the value of $(2 \hat{a}-\hat{b}) \cdot[(\hat{a} \times \hat{b}) \times(\hat{a}+2 \hat{b})]$ is

If the volume of a parallelepiped whose coterminous edges are represented by the vectors $-12i + \alpha k$,$3j - k$,and $2i + j - 15k$ is $546$,find the value of $\alpha$.