If $\vec{p}$ and $\vec{q}$ are unequal unit vectors such that $(\vec{p} - \vec{q}) \cdot ((2\vec{q} + \vec{p}) \times (3\vec{p} - \vec{q})) = |\vec{p} + \vec{q}|$,then the angle between $\vec{p}$ and $\vec{q}$ is:

If the points $A(2,1,-1), B(0,-1,0), C(4,0,4)$ and $D(2,0,x)$ are coplanar,then $x=$

If the vectors $\vec{a} = \hat{i} + a\hat{j} + \hat{k}$,$\vec{b} = \hat{j} + a\hat{k}$,and $\vec{c} = a\hat{i} + \hat{k}$ are given,find the value of $a$ for which the volume of the parallelepiped formed by these three vectors as coterminous edges is minimum.

Let $\vec{a}=\hat{i}+2\hat{j}+3\hat{k}$ and $\vec{b}=\hat{i}+\hat{j}-\hat{k}$. If $\vec{c}$ is a vector such that $\vec{a} \cdot \vec{c}=11$,$\vec{b} \cdot(\vec{a} \times \vec{c})=27$ and $\vec{b} \cdot \vec{c}=-\sqrt{3}|\vec{b}|$,then $|\vec{a} \times \vec{c}|^2$ is equal to

If the vectors $2i - j + k$,$i + 2j - 3k$,and $3i + \lambda j + 5k$ are coplanar,then $\lambda = $

If $\vec p$ and $\vec q$ are unit vectors such that $[\vec p, \vec q, \vec p \times \vec q] = \frac{1}{2}$,then the angle between $\vec p$ and $\vec q$ is