Let $\vec{a}=\hat{i}+\hat{j}+\hat{k}$, $\vec{b}=\hat{i}-2\hat{j}+\hat{k}$, $\vec{c}=\hat{i}+3\hat{j}-2\hat{k}$, and $\vec{d}=2\hat{i}+\hat{j}-\hat{k}$ be four vectors. Let $l=\vec{b} \cdot \vec{c}$ and $m=\vec{b} \cdot \vec{a}$. Find the value of the scalar triple product $[(m\vec{b}+l\vec{a}) \quad \vec{b} \quad \vec{d}]$.

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 $a=x \hat{i}+y \hat{j}+z \hat{k}$, then $(a \times \hat{i}) \cdot(\hat{i}+\hat{j})+(a \times \hat{j}) \cdot(\hat{j}+\hat{k})+(a \times \hat{k}) \cdot(\hat{k}+\hat{i})=$

Observe the following statements:
$A$. Three vectors are coplanar if one of them is expressible as a linear combination of the other two.
$R$. Any three coplanar vectors are linearly dependent.
Then, which of the following is true?

Volume of a parallelepiped with coterminous edges $\vec{a}, \vec{b}, \vec{c}$ is $12$ cubic units. The volume of a tetrahedron with coterminous edges $\vec{a} - \vec{b}, \vec{b} - \vec{c}, \vec{a} + \vec{b} - \vec{c}$ will be ............. cubic units.

Let $\overrightarrow{a}=\hat{i}-2 \hat{j}+3 \hat{k}$, $\overrightarrow{b}=2 \hat{i}+3 \hat{j}-\hat{k}$ and $\overrightarrow{c}=\lambda \hat{i}+\hat{j}+(2 \lambda-1) \hat{k}$. If $\overrightarrow{c}$ is parallel to the plane containing $\overrightarrow{a}$ and $\overrightarrow{b}$, then $\lambda$ is equal to