If $G(\bar{g})$,$H(\bar{h})$,and $P(\bar{p})$ are respectively the centroid,orthocenter,and circumcentre of a triangle and $x \bar{p} + y \bar{h} + z \bar{g} = \overline{0}$,then $x, y, z$ are respectively:

If the moduli of $a$ and $b$ are equal and the angle between them is $120^\circ$ and $a \cdot b = -8$,then $|a|$ is equal to

If $\vec{a}$,$\vec{b}$,and $\vec{a}-\vec{b}$ are unit vectors and the angle between the two vectors $\vec{a}$ and $\vec{b}$ is $\theta$,then $\theta = $ . . . . . . .

If $a, b$ and $c$ are unit vectors such that $a + b - c = 0,$ then the angle between $a$ and $b$ is

Given three vectors $a=2 \hat{i}-\hat{j}+\hat{k}$,$b=\hat{i}+2 \hat{j}-\hat{k}$,and $c=\hat{i}+\hat{j}-2 \hat{k}$,a vector in the plane of $b$ and $c$ whose projection on $a$ is of magnitude $\sqrt{\frac{2}{3}}$ is

For $p > 0$,a vector $\vec{v}_{2} = 2 \hat{i} + (p + 1) \hat{j}$ is obtained by rotating the vector $\vec{v}_{1} = \sqrt{3} p \hat{i} + \hat{j}$ by an angle $\theta$ about the origin in the counter-clockwise direction. If $\tan \theta = \frac{(\alpha \sqrt{3} - 2)}{4 \sqrt{3} + 3}$,then the value of $\alpha$ is equal to $....$