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

Let the position vectors of the vertices of a triangle $ABC$ be $\bar{a}, \bar{b}, \bar{c}$. If on the plane of the triangle,$P$ is a point having position vector $\bar{x}$ such that $\bar{x} \cdot (\bar{c} - \bar{b}) = \bar{a} \cdot \bar{c} - \bar{a} \cdot \bar{b}$ and $\bar{x} \cdot (\bar{a} - \bar{c}) = \bar{a} \cdot \bar{b} - \bar{b} \cdot \bar{c}$,then for the triangle $ABC$,$P$ is the

$a = 4 \hat{i} + 3 \hat{j}$ and $b$ are two vectors in the $XOY$ plane,and $a$ is perpendicular to $b$. $A$ vector $c$ lying in the same plane and having projections $1$ and $2$ on $a$ and $b$ respectively is:

If the angle between the two vectors $\vec{u} = \hat{i} + \hat{k}$ and $\vec{v} = \hat{i} - \hat{j} + a\hat{k}$ is $\pi/3$,find the value of $a$.

The vector $\vec{a} = \alpha \hat{i} + 2\hat{j} + \beta \hat{k}$ lies in the plane of $\vec{b} = \hat{i} + \hat{j}$ and $\vec{c} = \hat{j} + \hat{k}$ and bisects the angle between $\vec{b}$ and $\vec{c}$. Find the possible values of $\alpha$ and $\beta$.

Three vectors $\vec{a}, \vec{b}$ and $\vec{c}$ satisfy the condition $\vec{a}+\vec{b}+\vec{c}=\vec{0}.$ Evaluate the quantity $\mu=\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a},$ if $|\vec{a}|=1, |\vec{b}|=4$ and $|\vec{c}|=2.$