If the scalar triple product of the vectors $-3 \hat{i}+7 \hat{j}-3 \hat{k}$,$3 \hat{i}-7 \hat{j}+\lambda \hat{k}$ and $7 \hat{i}-5 \hat{j}-3 \hat{k}$ is $272$,then $\lambda = \ldots$

Let $O$ be the origin. Let $\overline{OP} = x\hat{i} + y\hat{j} - \hat{k}$ and $\overline{OQ} = -\hat{i} + 2\hat{j} + 3x\hat{k}$,where $x, y \in \mathbb{R}$ and $x > 0$,be such that $|\overline{PQ}| = \sqrt{20}$ and the vector $\overline{OP}$ is perpendicular to $\overline{OQ}$. If $\overline{OR} = 3\hat{i} + z\hat{j} - 7\hat{k}$,where $z \in \mathbb{R}$,is coplanar with $\overline{OP}$ and $\overline{OQ}$,then the value of $x^2 + y^2 + z^2$ is equal to ...... .

If $\vec{a}, \vec{b}, \vec{c}$ are non-coplanar vectors and $\lambda$ is a real number,then for what value of $\lambda$ are the vectors $\vec{a} + 2\vec{b} + 3\vec{c}$,$\lambda\vec{b} + 4\vec{c}$,and $(2\lambda - 1)\vec{c}$ non-coplanar?

The volume of a tetrahedron whose vertices are $4 \hat{i}+5 \hat{j}+\hat{k}$, $-\hat{j}+\hat{k}$, $3 \hat{i}+9 \hat{j}+4 \hat{k}$ and $-2 \hat{i}+4 \hat{j}+4 \hat{k}$ is (in cubic units)

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 $\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k}$ and $\overrightarrow{b}=\hat{j}-\hat{k}.$ If $\overrightarrow{c}$ is a vector such that $\vec{a} \times \vec{c}=\vec{b}$ and $\vec{a} \cdot \vec{c}=3$,then $\vec{a} \cdot(\vec{b} \times \vec{c})$ is equal to :