If $\alpha (a \times b) + \beta (b \times c) + \gamma (c \times a) = 0$ and at least one of the numbers $\alpha, \beta,$ and $\gamma$ is non-zero,then the vectors $a, b,$ and $c$ are

If $a(\alpha \times \beta)+b(\beta \times \gamma)+c(\gamma \times \alpha)=0$ and at least one of the scalars $a, b, c$ is non-zero,then the vectors $\alpha, \beta, \gamma$ are

$[\hat{i}-\hat{j}, \hat{j}-\hat{k}, \hat{k}-\hat{i}]$ is equal to

The value of $a$ for which the volume of the parallelepiped formed by $\hat{i} + a \hat{j} + \hat{k}$,$\hat{j} + a \hat{k}$ and $a \hat{i} + \hat{k}$ becomes minimum is

Let $\bar{a}=\lambda \bar{i}+3 \bar{j}+4 \bar{k}$, $\bar{b}=3 \bar{i}-\bar{j}+\lambda \bar{k}$ and $\bar{c}=\lambda \bar{i}+\bar{j}-3 \bar{k}$ be three vectors for some integer $\lambda$. If the volume of the parallelepiped with $\bar{a}, \bar{b}, \bar{c}$ as coterminus edges is $61$ cubic units, then the number of possible values of $\lambda$ is

The volume (in cubic units) of the tetrahedron with edges $\hat{i}+\hat{j}+\hat{k}$, $\hat{i}-\hat{j}+\hat{k}$ and $\hat{i}+2\hat{j}-\hat{k}$ is