If $\bar{a}=\hat{i}+\hat{j}+\hat{k}$,$\bar{b}=4\hat{i}+3\hat{j}+4\hat{k}$,and $\bar{c}=\hat{i}+\alpha\hat{j}+\beta\hat{k}$ are linearly dependent vectors and $|\bar{c}|=\sqrt{3}$,then the values of $\alpha$ and $\beta$ are respectively.

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three non-zero non-coplanar vectors. Let the position vectors of four points $A, B, C$ and $D$ be $\vec{a}-\vec{b}+\vec{c}$,$\lambda \vec{a}-3 \vec{b}+4 \vec{c}$,$-\vec{a}+2 \vec{b}-3 \vec{c}$ and $2 \vec{a}-4 \vec{b}+6 \vec{c}$ respectively. If $\overrightarrow{AB}$,$\overrightarrow{AC}$ and $\overrightarrow{AD}$ are coplanar,then $\lambda$ is :

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 $OA, OB, OC$ be the co-terminal edges of a rectangular parallelopiped of volume $V$ and let $P$ be the vertex opposite to $O$. Then,$[\overrightarrow{AP} \overrightarrow{BP} \overrightarrow{CP}]$ is equal to

Given vectors $a, b, c$ such that $a \cdot (b \times c) = \lambda \neq 0$,the value of $\frac{(b \times c) \cdot (a + b + c)}{\lambda}$ is

If the volume of the parallelepiped whose coterminous edges are along the vectors $\bar{a}, \bar{b}, \bar{c}$ is $12$,then the volume of the tetrahedron whose coterminous edges are $\bar{a}+\bar{b}, \bar{b}+\bar{c}$ and $\bar{c}+\bar{a}$ is