If the vectors $2i - j + k$,$i + 2j - 3k$,and $3i + \lambda j + 5k$ are coplanar,then $\lambda = $

The volume of a parallelepiped whose coterminous edges are represented by the vectors $\overrightarrow{OA} = (2, 1, 1)$,$\overrightarrow{OB} = (3, -1, 1)$,and $\overrightarrow{OC} = (-1, 1, -1)$ is . . . . . . cubic units.

If the vectors $a \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+b \hat{j}+\hat{k}$,and $\hat{i}+\hat{j}+c \hat{k}$ are coplanar,where $(a, b, c \neq 1)$,then the value of $\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=$

Which of the following expressions is meaningful?

Let $\vec{v}=\alpha \hat{i}+2 \hat{j}-3 \hat{k}$,$\vec{w}=2 \alpha \hat{i}+\hat{j}-\hat{k}$,and $\vec{u}$ be a vector such that $|\vec{u}|=\alpha > 0$. If the minimum value of the scalar triple product $[\vec{u} \vec{v} \vec{w}]$ is $-\alpha \sqrt{3401}$,and $|\vec{u} \cdot \hat{i}|^2=\frac{m}{n}$ where $m$ and $n$ are coprime natural numbers,then $m + n$ is equal to $.........$.

The vector $c \cdot (b+c) \times (a+b+c)$ is equal to