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 $\vec z$.

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-coplanar unit vectors such that the angle between every pair of them is $\frac{\pi}{3}$. If $\vec{a} \times \vec{b} + \vec{b} \times \vec{c} = p \vec{a} + q \vec{b} + r \vec{c}$,where $p, q$ and $r$ are scalars,then the value of $\frac{p^2 + 2q^2 + r^2}{q^2}$ is

The sum of all real values of $\lambda$ for which the vectors $\vec{a} = \lambda \hat{i} + \hat{j} + \hat{k}$, $\vec{b} = \hat{i} + \lambda \hat{j} + 2\hat{k}$, and $\vec{c} = 2\hat{i} + 3\hat{j} + \lambda \hat{k}$ are coplanar is:

Let $a, b, c$ be distinct non-negative numbers. If the vectors $a\hat{i} + a\hat{j} + c\hat{k}$,$\hat{i} + \hat{k}$,and $c\hat{i} + c\hat{j} + b\hat{k}$ lie in a plane,then $c$ is

If $\bar{a}=3 \hat{i}+\hat{j}-\hat{k}, \bar{b}=2 \hat{i}-\hat{j}+23 \hat{k}$ and $\bar{c}=7 \hat{i}-\hat{j}+23 \hat{k}$,then which of the following is valid?

If the volume of a tetrahedron having $\bar{i}+2 \bar{j}-3 \bar{k}$, $2 \bar{i}+\bar{j}-3 \bar{k}$, and $3 \bar{i}-\bar{j}+p \bar{k}$ as its coterminous edges is $2$, then the values of $p$ are the roots of the equation