If $P=(0,1,2), Q=(4,-2,1)$ and $O=(0,0,0)$,then $\angle POQ=$

The value of $\lambda$ for which the vectors $\vec{a} = 2\hat{i} + \lambda\hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2\hat{j} + 3\hat{k}$ are orthogonal is

In a triangle $ABC$, $D$ and $E$ divide the sides $BC$ and $CA$ in the ratio $2:1$ respectively. If $P$ is the point of intersection of $AD$ and $BE$, then the ratio in which $P$ divides $AD$ is

The set of real values of $\lambda$ for which the vectors $\lambda \hat{i}-3 \hat{j}+5 \hat{k}$ and $2 \lambda \hat{i}-\lambda \hat{j}+\hat{k}$ are perpendicular to each other is

$a, b, c$ are three vectors such that $|a|=3, |b|=5, |c|=7$. If $a, b, c$ are perpendicular to the vectors $b+c, c+a, a+b$ respectively,then $\sqrt{(a+b+c)^2-2}=$

If $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are unit vectors such that $(\vec{a} \times \vec{b}) \cdot (\vec{c} \times \vec{d}) = 1$ and $\vec{a} \cdot \vec{c} = \frac{1}{2}$,then: