If $a, b$ and $c$ are unit vectors such that $a+b+c=0$,then $a \cdot b+b \cdot c+c \cdot a$ is equal to

Let $\bar{a}, \bar{b}, \bar{c}$ be three unit vectors satisfying $|\bar{a}-\bar{b}|^2+|\bar{a}-\bar{c}|^2=10$. Then
Statement $(I)$ : $|\bar{a}+2 \bar{b}|^2+|2 \bar{a}+\bar{c}|^2=2$.
Statement $(II)$ : $|2 \bar{a}+3 \bar{b}|^2+|3 \bar{a}+2 \bar{c}|^2=10$.
Which of the above statements is (are) true?

Let $\vec{a} = 2\hat{i} + \lambda_{1}\hat{j} + 3\hat{k}$,$\vec{b} = 4\hat{i} + (3 - \lambda_{2})\hat{j} + 6\hat{k}$,and $\vec{c} = 3\hat{i} + 6\hat{j} + (\lambda_{3} - 1)\hat{k}$ be three vectors such that $\vec{b} = 2\vec{a}$ and $\vec{a}$ is perpendicular to $\vec{c}$. Then a possible value of $(\lambda_{1}, \lambda_{2}, \lambda_{3})$ is

The value of $x$ for which the angle between the vectors $a = -3i + xj + k$ and $b = xi + 2xj + k$ is acute and the angle between $b$ and the $x$-axis lies between $\pi/2$ and $\pi$ satisfies:

Let $\overline{a}, \overline{b}$ and $\overline{c}$ be vectors of magnitude $2, 3$ and $4$ respectively. If $\overline{a}$ is perpendicular to $(\overline{b}+\overline{c})$,$\overline{b}$ is perpendicular to $(\overline{c}+\overline{a})$ and $\overline{c}$ is perpendicular to $(\overline{a}+\overline{b})$,then the magnitude of $\overline{a}+\overline{b}+\overline{c}$ is equal to

If the vectors $\vec{a} = \hat{i} - 2x\hat{j} - 3y\hat{k}$ and $\vec{b} = \hat{i} + 3x\hat{j} + 2y\hat{k}$ are perpendicular to each other,find the locus of the point $(x, y)$.