The value of $a$ for which the volume of the parallelepiped formed by $\hat{i} + a \hat{j} + \hat{k}$,$\hat{j} + a \hat{k}$ and $a \hat{i} + \hat{k}$ becomes minimum is

If $\overline{a}$ and $\overline{c}$ are unit vectors inclined at $\frac{\pi}{3}$ with each other and $(\overline{a} \times (\overline{b} \times \overline{c})) \cdot (\overline{a} \times \overline{c}) = 5$,then the value of $5[\overline{a} \overline{b} \overline{c}] = $

If $(1,5,35), (7,5,5), (1, \lambda, 7)$ and $(2 \lambda, 1, 2)$ are coplanar,then the sum of all possible values of $\lambda$ is

Let $\vec{\lambda} = x\vec{a} + y\vec{b} + z\vec{c}$ and $\vec{\lambda} \cdot (\vec{a} \times \vec{b} + \vec{b} \times \vec{c} + \vec{c} \times \vec{a}) = 2(x + y + z)$ (where $x + y + z \neq 0$),then the scalar triple product $[\vec{a} \, \vec{b} \, \vec{c}]$ is:

The scalar $\overline{a} \cdot [(\overline{b} + \overline{c}) \times (\overline{a} + \overline{b} + \overline{c})]$ equals

$A$ unit vector coplanar with $i+j+3k$ and $i+3j+k$ and perpendicular to $i+j+k$ is