Let $\vec{a} = 2\hat{i} + 3\hat{j} - \hat{k}$ and $\vec{b} = \hat{i} - 2\hat{j} + 3\hat{k}$. For what value of $\lambda$ is the vector $\vec{c} = \lambda\hat{i} + \hat{j} + (2\lambda - 1)\hat{k}$ parallel to the plane containing $\vec{a}$ and $\vec{b}$?

If the points whose position vectors are $3i - 2j - k,$ $2i + 3j - 4k,$ $-i + j + 2k,$ and $4i + 5j + \lambda k$ lie on a plane,then $\lambda = $

If $a, b$ and $c$ are three non-coplanar vectors,then $(a + b + c) \cdot [(a + b) \times (a + c)]$ is equal to

If $\bar{a} = \bar{i} - \bar{j}$,$\bar{b} = \bar{j} - \bar{k}$,$\bar{c} = \bar{k} - \bar{i}$ and $\bar{d}$ is a unit vector such that $\bar{a} \cdot \bar{d} = 0$ and $[\bar{b} \bar{c} \bar{d}] = 0$,then the vector $\bar{d} = ....$

The value of $\frac{[(\vec{a} \times \vec{b}) \times (\vec{b} \times \vec{c}), (\vec{b} \times \vec{c}) \times (\vec{c} \times \vec{a}), (\vec{c} \times \vec{a}) \times (\vec{a} \times \vec{b})]}{[\vec{a} \times \vec{b}, \vec{b} \times \vec{c}, \vec{c} \times \vec{a}]}$ is equal to

The volume of the parallelepiped formed by the vectors $\hat{i} + m \hat{j} + \hat{k}$,$\hat{j} + m \hat{k}$,and $m \hat{i} + \hat{k}$ becomes minimum when $m$ is