If $a, b, c$ are distinct non-negative numbers and the vectors $a\hat{i} + a\hat{j} + c\hat{k}$, $\hat{i} + \hat{k}$, and $c\hat{i} + c\hat{j} + b\hat{k}$ lie in the same plane, then the value of $c$ is...

If the vectors $\vec{a}=2 \hat{i}-\hat{j}+\hat{k}$,$\vec{b}=\hat{i}+2 \hat{j}-3 \hat{k}$,and $\vec{c}=3 \hat{i}+p \hat{j}+5 \hat{k}$ are coplanar,then $p=$

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 ...... .

The value of $[\vec{a}-\vec{b} \quad \vec{b}-\vec{c} \quad \vec{c}-\vec{a}]$ is equal to

If $\vec{u} = \hat{j} + 4\hat{k}$,$\vec{v} = \hat{i} + 3\hat{k}$ and $\vec{w} = \cos \theta \hat{i} + \sin \theta \hat{j}$ are vectors in $3$-dimensional space,then the maximum possible value of $|(\vec{u} \times \vec{v}) \cdot \vec{w}|$ is

The volume of a tetrahedron whose vertices are $A \equiv (-1, 2, 3)$,$B \equiv (3, -2, 1)$,$C \equiv (2, 1, 3)$,and $D \equiv (-1, -2, 4)$ is