If the vectors $\bar{a}=\hat{\imath}-2 \hat{\jmath}+\hat{k}$,$\bar{b}=2 \hat{\imath}-5 \hat{\jmath}+p \hat{k}$ and $\bar{c}=5 \hat{\imath}-9 \hat{\jmath}+4 \hat{k}$ are coplanar,then the value of $p$ is

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?

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

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

$\vec{a}=2 \hat{i}-\hat{j}$,$\vec{b}=2 \hat{j}-\hat{k}$,$\vec{c}=2 \hat{k}-\hat{i}$ are three vectors and $\vec{d}$ is a unit vector perpendicular to $\vec{c}$. If $\vec{a}, \vec{b}, \vec{d}$ are coplanar vectors,then $|\vec{d} \cdot \vec{b}|=$

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