Given the vectors $\vec x = 3i - 6j - k$,$\vec y = i + 4j - 3k$,and $\vec z = 3i - 4j - 12k$,find the projection of the vector $\vec x \times \vec y$ onto the vector $\vec z$.

Value of $\hat{j} \cdot(\hat{i} \times \hat{k})+\hat{i} \cdot(\hat{j} \times \hat{j})+\hat{k} \cdot(\hat{j} \times \hat{i})+\hat{i} \cdot(\hat{k} \times \hat{j})$ is . . . . . . .

$a$ is perpendicular to both $b$ and $c$. The angle between $b$ and $c$ is $\frac{2 \pi}{3}$. If $|a|=2$, $|b|=3$, and $|c|=4$, then $c \cdot (a \times b)$ is equal to (in $\sqrt{3}$)

If the volume of the parallelopiped with $\vec{a} \times \vec{b}, \vec{b} \times \vec{c}$ and $\vec{c} \times \vec{a}$ as coterminous edges is $9 \text{ cu. units}$, then the volume of the parallelopiped with $(\vec{a} \times \vec{b}) \times(\vec{b} \times \vec{c}),(\vec{b} \times \vec{c}) \times(\vec{c} \times \vec{a})$ and $(\vec{c} \times \vec{a}) \times(\vec{a} \times \vec{b})$ as coterminous edges is

If the vectors $p \hat{i}+\hat{j}+\hat{k}$,$\hat{i}+q \hat{j}+\hat{k}$ and $\hat{i}+\hat{j}+r \hat{k}$ $(p \neq q \neq r \neq 1)$ are coplanar,then the value of $pqr-(p+q+r)$ 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