If $\overrightarrow{a} \cdot \hat{i}=4$,then $(\overrightarrow{a} \times \hat{j}) \cdot(2 \hat{j}-3 \hat{k})$ is equal to

Let $a = 2i + j + k$ and $b = i + 2j - k$. If a unit vector $c$ is coplanar with $a$ and $b$,and $c$ is perpendicular to $a$,then $c$ is:

The cosine of the angle $A$ of the triangle with vertices $A(1, -1, 2)$,$B(6, 11, 2)$,and $C(1, 2, 6)$ is:

If $\bar{a}$ and $\bar{b}$ are vectors such that $|\bar{a}+\bar{b}|=\sqrt{29}$ and $\bar{a} \times(2 \hat{i}+3 \hat{j}+4 \hat{k})=(2 \hat{i}+3 \hat{j}+4 \hat{k}) \times \bar{b}$,then a possible value of $(\bar{a}+\bar{b}) \cdot(-7 \hat{i}+2 \hat{j}+3 \hat{k})$ is

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

Let $P$ be a real number and $|P| \geq 2$. If $A, B, C$ are variable angles such that $(\sqrt{P^2-4}) \tan A + P \tan B + (\sqrt{P^2+4}) \tan C = 6P$,then the minimum value of $\tan^2 A + \tan^2 B + \tan^2 C$ is: