If three vectors $2\hat{i}-\hat{j}-\hat{k}$,$\hat{i}+2\hat{j}-3\hat{k}$ and $3\hat{i}+\lambda\hat{j}+5\hat{k}$ are coplanar,then the value of $\lambda$ is

$i \cdot (j \times k) + j \cdot (k \times i) + k \cdot (i \times j) = $

If $a=2 \hat{i}-3 \hat{j}+5 \hat{k}$, $b=3 \hat{i}-4 \hat{j}+5 \hat{k}$ and $c=5 \hat{i}-3 \hat{j}-2 \hat{k}$, then the volume of the parallelepiped with coterminous edges $a+b$, $b+c$, $c+a$ is

The volume of the parallelepiped whose edges are represented by $-12i + \alpha k$,$3j - k$,and $2i + j - 15k$ is $546$. Then $\alpha = $

The value of $(\vec{a} + 2\vec{b} - \vec{c}) \cdot \{(\vec{a} - \vec{b}) \times (\vec{a} - \vec{b} - \vec{c})\}$ is equal to

Let $a = i - k$, $b = xi + j + (1 - x)k$, and $c = yi + xj + (1 + x - y)k$. Then $[a\,b\,c]$ depends on