Three lines are drawn from the origin $O$ with direction ratios proportional to $(1, -1, 1)$, $(2, -3, 0)$, and $(1, 0, 3)$. The three lines are

Let $\overline{a} = \hat{i} + \hat{j} + \hat{k}$,$\overline{b} = \hat{i} - \hat{j} + 2\hat{k}$,and $\overline{c} = x\hat{i} + (x - 2)\hat{j} - \hat{k}$. If the vector $\overline{c}$ lies in the plane of $\overline{a}$ and $\overline{b}$,then $x = \dots$

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:

If $\alpha (a \times b) + \beta (b \times c) + \gamma (c \times a) = 0$ and at least one of the numbers $\alpha, \beta,$ and $\gamma$ is non-zero,then the vectors $a, b,$ and $c$ are

If $a, b, c$ are non-coplanar vectors and $\lambda$ is a real number,then the vectors $a + 2b + 3c, \lambda b + 4c$ and $(2\lambda - 1)c$ are non-coplanar for

If $\vec{\alpha}$ is a unit vector, $\vec{\beta}=\hat{i}+\hat{j}-\hat{k}$, and $\vec{\gamma}=\hat{i}+\hat{k}$, then the maximum value of $[\vec{\alpha} \vec{\beta} \vec{\gamma}]$ is