If $\bar{a} = \hat{i} + \hat{j} + \hat{k}, \bar{b} = \hat{i}, \bar{c} = c_1\hat{i} + c_2\hat{j} + c_3\hat{k}$ with $c_1 = 1$, $c_2 = 2$, then value of $c_3$ such that $\bar{a}, \bar{b}, \bar{c}$ are coplanar is ____

If $\bar{a}, \bar{b}$ and $\bar{c}$ are non-coplanar unit vectors such that the angle between any two of them is $60^\circ$, and the vector $\bar{d} = x\bar{a} + y\bar{b} + z\bar{c}$ is perpendicular to both $\bar{a}$ and $\bar{b}$, then the value of $\dfrac{(x+y)}{z}$ is

If $\bar{a}, \bar{b}, \bar{c}$ are three non zero and non-coplanar vectors such that $\bar{a} \times (\bar{b} \times \bar{c}) = \dfrac{\bar{b}}{2}$, then the angle between $\bar{a}$ and $\bar{b}$ is ...

The vector equation of the plane which is at a distance of $5$ units from the origin and normal to the vector $2\hat{i} + \hat{j} - 2\hat{k}$ is