If a vector $\alpha$ lies in the plane of $\beta$ and $\gamma$,then which of the following is correct?

Statement-$1$: Vectors $\vec{a}, \vec{b},$ and $\vec{c}$ are coplanar if and only if $\vec{a} \cdot (\vec{b} \times \vec{c}) = 0$.
Statement-$2$: Vectors $\vec{u}$ and $\vec{v}$ are perpendicular if and only if $\vec{u} \cdot \vec{v} = 0$,where $\vec{u} \times \vec{v}$ is a vector perpendicular to the plane of $\vec{u}$ and $\vec{v}$.

The volume of the tetrahedron having vertices $(1, -6, 10)$,$(-1, -3, 7)$,$(5, -1, \lambda)$ and $(7, -4, 7)$ is $11 \text{ cubic units}$. Then $\lambda = $

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

Let $\vec{a} = \hat{i} - 2\hat{j} + 3\hat{k}$,$\vec{b} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{c}$ be a vector such that $\vec{a} \times \vec{c} = \vec{b}$ and $\vec{a} \cdot \vec{c} = 3$. If $\vec{c} = x\vec{a} + y\vec{b} + z(\vec{a} \times \vec{b})$,then the value of $x + y + z$ is:

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