If the four points $A(6,2,4)$,$B(1,3,5)$,$C(1,-2,3)$,and $D(6, k, 2)$ are coplanar,then $k=$

If $\bar{a}=\hat{i}+\hat{j}+\hat{k}$,$\bar{b}=4\hat{i}+3\hat{j}+4\hat{k}$,and $\bar{c}=\hat{i}+\alpha\hat{j}+\beta\hat{k}$ are linearly dependent vectors and $|\bar{c}|=\sqrt{3}$,then the values of $\alpha$ and $\beta$ are respectively.

If $a, b, c$ are any three non-coplanar vectors,then $[a + b, b + c, c + a] = $

The value of $a$ such that the volume of the parallelepiped formed by the vectors $i + aj + k$,$j + ak$,and $ai + k$ is minimum is:

The volume of the parallelopiped whose coterminous edges are $\hat{j}+\hat{k}$, $\hat{i}+\hat{k}$, and $\hat{i}+\hat{j}$ is

If the scalar triple product of the vectors $-3 \hat{i}+7 \hat{j}-3 \hat{k}$,$3 \hat{i}-7 \hat{j}+\lambda \hat{k}$ and $7 \hat{i}-5 \hat{j}-3 \hat{k}$ is $272$,then $\lambda = \ldots$